behavior

JoVE Behavior documents observational and experimental techniques that seek to understand human and animal behavior. Work in this field includes analysis of genetic causes and physiological responses to internal or external stimuli as well as the implicated brain regions.

Behavior

biochemistry

JoVE Biochemistry publishes experiments that investigate the structure, function, and interactions of biological molecules. The video articles in this section demonstrate methods to study nucleic acids, proteins, carbohydrates and lipids as well as enzyme function, protein folding, molecular interactions and biochemical reactions.

Biochemistry

bioengineering

JoVE Bioengineering merges both physical and life sciences to understand and predict biological processes. Applying physical science tools to life science questions allow for the discovery of better technologies to measure, diagnose, and clinically treat disease.

Bioengineering

biology

JoVE Biology welcomes all general biology research methodologies. Content in this section canvases all fields of cell, molecular, and organismal biology, ranging from new applications of standard techniques to novel approaches aimed at understanding the functions of life and living organisms. This diverse section includes, but is not limited to, techniques in physical biology, cellular biochemistry, genetics, physiology, systems biology and a combination of eukaryotic and prokaryotic model systems.

Biology

cancer research

JoVE Cancer Research illustrates experimental approaches in biomedical research and clinical practice aimed at understanding, detecting, treating and preventing cancer. The video articles in this section demonstrate methods to study the cellular and molecular mechanisms of cancer, animal models of cancer research, development of new therapies and diagnostics, tumor imaging, chemical and radiation treatments, surgical procedures and improvements in therapy. These video articles offer researchers and clinical practitioners an efficient way to learn the technologies and experimental methods in cancer research.

Cancer Research

chemistry

JoVE Chemistry surveys a broad scope of chemical disciplines. This collection includes fundamental and applied research that investigates molecular interactions, properties, and structure spanning in the following core divisions: analytical, organic, organometallic, inorganic, bioinorganic, and biochemistry. This section also includes the design and preparation of materials and small molecules for advanced or unique applications.

Chemistry

developmental biology

JoVE Developmental Biology publishes research methodologies in the field of developmental biology. This includes methods that are used to study biological development in vitro and in vivo at the molecular, cellular, tissue, organ and whole organism levels.

Developmental Biology

engineering

JoVE Engineering encompasses a broad range of experimental and instrumental techniques utilized in physics research. Investigations in this area strive to address and answer a broad range of scientific questions, such as device mechanisms and efficiencies, using physical tools. This approach often requires a combination of specialties, and research in this area tends to be interdisciplinary with contributions from mechanical, electrical, and chemical engineers.

Engineering

environment

JoVE Environment is dedicated to research methodologies that address environmental concerns and seek to better understand Earth's ecosystem.  Special consideration is given to experimental methodologies that assess society’s impact on the environment, suggest solutions for protecting Earth’s resources, and develop sustainable fuel sources. Moving towards an environmentally conscious planet is a collaborative effort with contributions from both life and physical scientists.

Environment

genetics

JoVE Genetics publishes video articles demonstrating experimental approaches to investigate gene function and regulation. This includes methods to study gene expression, mutagenesis, epigenetics, genetic determinants of development, genetic disorders, population genetics and evolution.

Genetics

immunology and infection

JoVE Immunology and Infection includes content from a variety of fields working toward a shared initiative of improving global health. Techniques evaluating the biological response to pathogens from the molecular to organismal level are presented as well as therapeutic agents and their efficacy in treating disease.

Immunology and Infection

medicine

JoVE Medicine serves as a connection between basic research and clinical applications. Articles may present medical procedures, case studies or clinical trial methodologies. Included studies are often modeled towards predictive, preventative, personalized, and practical approaches to treatment of disease. The goal of research methodologies in this section is to improve overall patient care and prognosis.

Medicine

neuroscience

JoVE Neuroscience is a multidisciplinary section devoted to investigations of the structure, function, physiology, and pathophysiology of the brain and nervous system. Included methodologies range from molecular and cellular level studies to full central and peripheral neural systems. Potential treatment platforms and surgical techniques for neurological diseases and disorders are also presented in this section.

Neuroscience

jove journal

JoVE Journal

Scientific video journal. Peer reviewed. Multi-disciplinary. Indexed in PubMed and Web of Science.

https://cloudfront.jove.com/files/banners/journal/journal_lander.jpg

Research

Pipe Flowrate Measurement: Problem Solving

A spray tank system is engineered to uniformly distribute a pest-control liquid across plants by using a pressurized mechanism. The tank, pressurized to 150 kPa, holds the pesticide at a height of 0.80 meters. Liquid flows from the tank through a 1.9 meter pipe with a diameter of 0.015 meters, angled at 0.698 radians, ultimately reaching a 0.007 meter nozzle that sprays the pesticide. Accurate calculation of the system&#39;s flow rate is crucial to ensure uniform application, and this is achieved by first determining the flow velocity within the connecting pipe.
The design requires an understanding of fluid mechanics principles. At the outset, the velocity of the fluid in the tank at Point 1 is zero, with the pressure set to 150 kPa. Point 2, located at the nozzle, is assumed to be open to the atmosphere, with a pressure of zero. The height difference between Points 1 and 2, calculated using the 0.698-radian pipe angle and pipe length, determines the elevation change component in the Bernoulli equation. This height difference is found by multiplying the pipe length by the sine of the inclination angle.
To connect the parameters at Points 1 and 2, a modified Bernoulli equation is used, accounting for energy losses due to friction and the nozzle&#39;s loss coefficient. The equation expresses the pressure, velocity, and elevation terms for both points as follows:
<img alt="Equation 1" src="/files/ftp_upload/18085/18085_PT_equation_1.svg" style="text-align: center; height: 30px;" />
Here, p represents pressure, &#961; the fluid density, V the velocity, g the gravitational acceleration, z1 the height of the pesticide inside the tank, and z2&#160; the nozzle height above the bottom of the tank. With values substituted for these terms, the flow velocity in the pipe at Point 1 is solved.
After calculating the pipe&#39;s flow velocity, the flow rate, Q, is determined by multiplying this velocity by the cross-sectional area A of the pipe:
<img alt="Equation 2" src="/files/ftp_upload/18085/18085_PT_equation_2.svg" style="text-align: center; height: 11px;" />
Where A is:
<img alt="Equation 3" src="/files/ftp_upload/18085/18085_PT_equation_3.svg" style="text-align: center; height: 25px;" />
This flow rate calculation ensures that the pesticide is delivered at a consistent and effective rate, optimizing pest control application.

Consider a spray tank system designed to evenly apply pest-control liquid to plants.
The system includes a pressurized tank at 150 kPa filled with ...

Pipe Flowrate Measurement

In pipe flow measurement, orifice, nozzle, and Venturi meters are commonly used to determine fluid flowrates by constricting the flow area, which increases fluid velocity and reduces pressure. This pressure difference, governed by Bernoulli&#39;s principle and adjusted for real-world conditions, is essential for calculating flowrate. Each meter type is suited to specific applications based on accuracy, efficiency, and compatibility with various flow conditions.
The orifice meter is a simple, cost-effective device consisting of a flat plate with a central hole that creates a measurable pressure drop. This device is popular in applications like water treatment, oil and gas processing, and general industrial use where moderate accuracy is acceptable. However, due to head loss from turbulent flow and area contraction, a discharge coefficient is used to correct the idealized flow equation. The flowrate through an orifice meter is proportional to the square root of the pressure difference, accounting for fluid density, orifice area, and the ratio of the hole diameter to the pipe diameter.
The nozzle meter improves on the orifice meter by using a contoured design that reduces flow separation and turbulence, resulting in a smoother flow and increased measurement accuracy. It is often used for high-velocity flows, such as in steam or gas pipelines, chemical processing, and power plants. With a higher discharge coefficient than the orifice meter, the nozzle meter provides improved precision with less energy loss.
The Venturi meter offers the highest accuracy among the three types. It is ideal for applications requiring minimal head loss and reliable measurements, such as water supply systems, natural gas pipelines, and wastewater management. Its smooth contraction and gradual expansion reduce flow separation, conserving energy and enabling accurate flowrate calculations even in low-pressure systems. The discharge coefficient for a Venturi meter, which depends on factors like the ratio of the throat to pipe diameter and flow characteristics, supports exact flow measurement but requires a more complex installation and is generally more costly.
Each meter type offers unique benefits, from essential flow monitoring to precise flow control, chosen based on the application&#39;s accuracy needs and specific conditions.

Flow measurement devices&#8212;such as orifice, nozzle, and Venturimeters&#8212;operate by narrowing the flow area. This increases fluid velocity and ...

Design Example: Designing a Residential Plumbing System

The design of residential plumbing systems requires carefully evaluating water demand, flow rates, and pressure dynamics to ensure both efficiency and reliability. The nature of water flow within pipes is defined by its Reynolds number, which classifies flow as either laminar (smooth) or turbulent.
<img alt="Equation 1" src="/files/ftp_upload/18086/18086_Equation_1.svg" style="text-align: center; height: 42px;" />
This dimensionless value depends on factors such as water velocity and pipe. Residential plumbing systems are designed to meet specific water demand and flow rate requirements based on occupancy and fixture needs.
Determining the nature of water flow &#8212; laminar or turbulent &#8212; requires calculating the Reynolds number, which depends on variables like pipe diameter and water velocity. In residential settings, flow is often turbulent, leading to higher energy loss due to friction, which impacts system efficiency. Energy losses, or head losses, arise as water moves through pipes. These losses are classified as either major or minor. Major losses from friction along pipe walls are calculated using the Darcy-Weisbach equation.
<img alt="Equation 2" src="/files/ftp_upload/18086/18086_Equation_2.svg" style="text-align: center; height: 42px;" />
The Darcy-Weisbach equation requires a friction factor, which is obtained from the Moody chart. This chart accounts for pipe roughness and flow characteristics, which significantly affect turbulent flow friction levels. Minor losses, in contrast, are caused by fittings, bends, and valves, each of which contributes resistance to flow. Each fitting is assigned a loss coefficient that quantifies its resistance and is factored into the total head loss.
<img alt="Equation 3" src="/files/ftp_upload/18086/18086_Equation_3.svg" style="text-align: center; height: 48px;" />
By adding major and minor losses, the overall head loss can be determined and managed so as to maintain adequate pressure, avoiding potential leaks or damage. Flow measurement devices like Venturi or orifice meters ensure the plumbing system provides the necessary flow rates. These meters measure flow rates by detecting pressure differences across constrictions in the pipe, confirming that water flow meets design specifications and household needs.

The residential plumbing system is designed by assessing the building&#39;s water demand and flow rate needs based on occupancy and fixture requirements.
...

Single Pipe Systems

In pipe flow analysis, problems are typically categorized into three types &#8212; Type I, Type II, and Type III &#8212; based on the known parameters and the desired outcome. Each type of problem addresses specific engineering requirements using fluid properties, pipe characteristics, and operational conditions.
In a Type I problem, fluid properties (density and viscosity), pipe characteristics (including diameter, length, and surface roughness), and the flow rate or average velocity are known. The objective is calculating the pressure drop or head loss needed to maintain the specified flow rate. For instance, in a heating system circulating hot water at 0.32 liters per second, knowing the system&#39;s fluid and pipe specifications allows engineers to determine the required pressure to achieve steady flow. Type I problems are common in designing HVAC systems, where maintaining consistent flow is essential for effective temperature control.
A Type II problem provides the applied pressure, fluid properties, and pipe details, to calculate the resulting flow rate. This type of analysis is used in applications such as garden irrigation, where engineers need to determine the water flow rate in a sprinkler system operating at a specified pressure, such as 276 kilopascals. Knowing the pipe dimensions and roughness, the flow rate reaching each sprinkler can be calculated to ensure adequate water distribution.
In a Type III problem, the known parameters include the pressure drop and desired flow rate, to determine the optimal pipe diameter to support efficient flow. This scenario is typical in cooling systems or industrial piping design, where engineers must size the pipe to achieve a flow rate of 0.19 liters per second, balancing efficiency with minimal energy loss. Choosing an appropriate pipe diameter in Type III problems ensures the required flow rate while minimizing pressure losses and pump energy demands.
By systematically categorizing pipe flow problems, engineers can streamline analysis and ensure optimal design and performance for fluid transport systems in heating, irrigation, and industrial applications.

In pipe flow problems, scenarios are grouped by known information and desired calculation and are typically classified as Type I, II, or III.
In a Type I ...

Minor Losses in Pipes

In pipe systems, minor losses refer to energy losses arising from components such as valves, bends, fittings, expansions, and other features that disrupt the steady flow of fluid. These disturbances cause energy dissipation through turbulence and resistance, which engineers quantify to manage system efficiency effectively.
Valves play a significant role in generating minor losses by obstructing or redirecting the fluid flow. When a valve is closed or partially closed, it restricts the flow path, creating additional resistance and inducing turbulence. The resulting energy dissipation depends on the valve&#39;s geometry, position, and flow path design. To quantify this effect, engineers use a loss coefficient, K, which scales with the square of the fluid velocity V as follows:
<img alt="Equation 1" src="/files/ftp_upload/18081/18081_Equation_1.svg" style="text-align: center; height: 30px;" />
where:
<ol style="list-style-type: circle;">
	<li>hLminor is the head loss due to the component,</li>
	<li>KL is the loss coefficient specific to the component geometry,</li>
	<li>V is the fluid velocity</li>
	<li>g is the acceleration due to gravity.</li>
</ol>
For ease of calculation, engineers often represent minor losses from fittings and valves as an equivalent length of straight pipe that would produce the same head loss.
At the entry point of a pipe, sharp edges or abrupt changes in flow area can cause the fluid to separate from the wall, creating turbulence and energy dissipation through viscous effects. This entrance loss contributes to the overall energy loss, depending on the pipe geometry and the flow velocity. When fluid exits a pipe, the kinetic energy disperses into the surrounding environment, creating an exit loss characterized by a unity loss coefficient, as all the kinetic energy is lost from the system.
Sudden expansions in the pipe create high-velocity jets that decelerate as they expand, dissipating energy through viscous effects. These expansions lead to additional energy losses as the fluid experiences abrupt pressure and velocity changes. Bends in the pipe also contribute to head loss, primarily due to flow separation and swirling motions induced by centripetal forces as the fluid changes direction. The frictional resistance over the length of the bend adds further to the overall head loss, with the degree of loss depending on the bend angle and radius.
Understanding these minor losses allows engineers to design pipe systems that manage energy dissipation effectively, optimizing flow efficiency and minimizing pressure drops across the system.

Minor losses in pipe systems arise from components like valves, bends, and fittings that disrupt fluid flow, causing energy dissipation through turbulence ...

Major Losses in Pipes

When a fluid flows through a pipe, it experiences energy losses due to frictional resistance along the pipe walls, known as major losses. These energy losses result in a pressure drop, which varies based on the flow conditions &#8212; whether laminar or turbulent &#8212; and the specific physical properties of the fluid and pipe.
Fluid flow can be classified as laminar or turbulent, primarily based on the Reynolds number. This dimensionless number reflects the relative influence of inertial to viscous forces in the fluid. In laminar flow (Re &#60; 2000), the fluid flows in parallel layers, or streamlines, with minimal cross-stream mixing. Here, the pressure drop mostly depends on the fluid&#39;s viscosity and is generally lower. In contrast, turbulent flow (Re &#62; 4000) is characterized by chaotic eddies and swirling motions. In this regime, the pressure drop is influenced not only by viscosity but also by the roughness of the pipe wall, as these irregularities disrupt the flow further, increasing energy losses.
The pressure drop in a pipe depends on several factors: fluid properties, flow velocity, pipe characteristics, and dimensionless numbers.
The Darcy-Weisbach equation is the standard approach for quantifying the pressure drop due to frictional losses in pipe flow:
<img alt="Equation 1" src="/files/ftp_upload/18080/18080_Equation_1.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>&#916;P is the pressure drop,</li>
	<li>f is the friction factor,</li>
	<li>L is the length of the pipe,</li>
	<li>D is the pipe diameter,</li>
	<li>&#961; is the fluid density,</li>
	<li>v is the flow velocity.</li>
</ol>
The friction factor f is crucial for calculating pressure drops, especially in turbulent flow. It depends on the Reynolds number and the relative roughness of the pipe. Engineers often refer to the Moody chart, which provides empirical values of friction factors across flow regimes and roughness levels.
The Colebrook equation
<img alt="Equation 2" src="/files/ftp_upload/18080/18080_Equation_2.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>f is the friction factor,</li>
	<li>&#1013;/D is the relative roughness</li>
	<li>Re is the Reynolds number.</li>
</ol>
offers a precise method to calculate f for smooth and moderately rough pipes, but it is implicit and requires iterative solutions. To avoid iteration, approximations like the Haaland equation:
<img alt="Equation 3" src="/files/ftp_upload/18080/18080_Equation_3.svg" style="text-align: center; height: 30px;" />
Where:
<ol style="list-style-type: circle;">
	<li>f is the friction factor,</li>
	<li>&#1013;/D is the relative roughness</li>
	<li>Re is the Reynolds number.</li>
</ol>
are commonly used in practice, providing reasonably accurate friction factor values without extensive calculation. Understanding these dynamics enables engineers to design pipe systems that manage flow efficiently, compensating for potential pressure losses over time as pipe roughness increases.

Consider &#160;a fluid flowing through a pipe; it encounters frictional resistance, causing energy losses known as major losses, resulting in a pressure ...

Multiple Pipe Systems

Multipipe systems consist of complex configurations of interconnected pipes designed to transport fluids efficiently across intricate networks. They are essential in engineering applications requiring precise control over flow distribution, pressure, and head loss. They are categorized into series, parallel, loop, and network configurations, each distinguished by unique flow characteristics and applications.
Series Configuration
In a series configuration, fluid flows sequentially from one pipe segment to the next, maintaining a constant flow rate throughout. This setup means each segment experiences the same flow rate, although head losses accumulate progressively across each section. The total head loss from an entry point A to an exit point B is the cumulative sum of head losses in each pipe, governed by the Darcy-Weisbach equation for head loss or similar empirical relationships. This configuration suits applications requiring a consistent flow rate along a fixed path, such as in specific water treatment processes.
Parallel Configuration
Parallel multipipe systems divide fluid flow across multiple paths, enabling a higher overall flow rate and reducing resistance across the network. Here, the total flow rate is the sum of the individual flow rates in each branch. However, a critical feature of parallel systems is that all paths experience the same pressure drop. Consequently, fluid particles in each path encounter identical head losses, regardless of the path taken. This arrangement is effective in systems needing efficient fluid distribution, such as in heating and cooling systems, where multiple paths ensure temperature control across large areas.
Loop and Network Configurations
In loop systems, the flow rate within one pipe is balanced by the combined flow rates of the other connected pipes, enabling fluid redistribution based on demand changes or localized resistance variations, such as in urban water distribution systems. Network configurations represent the most complex setup, involving numerous interconnected pipes with multiple inlets and outlets,such as in oil and gas pipeline networks. This flexibility allows for multiple flow directions, enhancing transport efficiency and adaptability to dynamic system demands.
Applications of Multipipe Systems
Multipipe systems are indispensable in several engineering sectors. Municipal water supply networks often use network configurations to ensure consistent pressure and reliable delivery across urban areas. In deepwater oil and gas pipelines, multipipe systems help manage the challenging pressures and flow conditions of underwater environments. Additionally, industrial cooling systems benefit from the controlled fluid distribution provided by parallel configurations, which improve heat dissipation and maintain optimal temperatures in manufacturing processes.

Multiple pipe systems consist of interconnected pipes that regulate fluid flow across complex networks.
They can be arranged in series, parallel, loop, or ...

Turbulent Flow

Turbulent flow is characterized by unpredictable fluctuations in velocity and pressure, which result in a chaotic fluid movement distinct from the orderly patterns of laminar flow. While laminar flow is governed by smooth, parallel layers with minimal mixing, turbulent flow exhibits highly irregular, three-dimensional patterns. This behavior arises due to instabilities in the fluid&#39;s velocity profile, and amplifies as the flow velocity increases. Minor disturbances, known as turbulent spots, appear once the flow reaches a critical speed. When the Reynolds number &#8212; a dimensionless quantity expressing the ratio of inertial to viscous forces in a fluid &#8212; exceeds a threshold of around 4,000, these spots coalesce, generating fully developed turbulence.
Structure of Turbulent Flow in Pipes
In pipe systems, turbulent flow comprises three distinct regions: the viscous sublayer, the buffer layer, and the outer layer. The viscous sublayer forms close to the pipe wall, a thin boundary where viscous forces dominate, and the flow behaves similarly to laminar flow. Here, velocity follows a near-linear profile influenced by fluid viscosity, providing a stable region within the otherwise chaotic turbulent flow.
Above the viscous sublayer lies the buffer layer, a transitional zone where viscous and turbulent forces contribute to fluid motion. This region is marked by a logarithmic velocity profile that conforms to the logarithmic law of the wall, representing a balance between viscous and turbulent interactions. Further from the wall, turbulent eddies drive fluid movement in the outer layer, promoting rapid mixing and random motion across the flow.
Engineering Applications of Turbulent Flow
The chaotic nature of turbulent flow makes it advantageous in numerous engineering applications where mixing and energy dissipation are crucial. For instance, wastewater treatment systems rely on turbulent flow to enhance the mixing of chemicals, promoting efficient contaminant removal. In hydraulic engineering, turbulent flow dissipates energy in spillways, preventing structural damage by controlling water velocity. The turbulent mixing process also reduces river sediment buildup, maintaining navigable waterways and protecting aquatic ecosystems.

Turbulent flow is a fluid movement marked by chaotic shifts in pressure and velocity.
Unlike laminar flow, turbulent flow displays complex, irregular ...

Turbulent Flow: Problem Solving

Carbonation is a process used to dissolve carbon dioxide gas in a liquid, commonly used in the production of carbonated beverages. Achieving efficient carbonation requires careful control of temperature, pressure, and flow conditions. By adjusting these parameters, carbonation efficiency can be maximized, producing a higher concentration of CO2&#160;in the liquid.
Temperature is a key factor in CO2 solubility. In this case, the CO2 gas and the liquid are cooled to 20&#176;C. Lower temperatures enhance the solubility of CO2, enabling a greater volume of gas to dissolve within the liquid. At this temperature, the CO2&#160;gas is pressurized to approximately 550 kPa. Elevated pressure increases the partial pressure of CO2&#160;over the liquid, further driving dissolution.
CO2 is pumped through a connecting pipe between two tanks, where the flow characteristics are crucial for maintaining steady carbonation. The high Reynolds number, defined as greater than 4000, indicates that the flow is turbulent rather than laminar. Turbulent flow improves gas-liquid contact by promoting mixing, which enhances CO2&#160;dissolution in the liquid. The Reynolds number (Re) is calculated using the equation:
<img alt="Equation 1" src="/files/ftp_upload/18079/18079_Equation_1.svg" style="text-align: center; height: 30px;" />
where:
<ol style="list-style-type: circle;">
	<li>&#961; is the density of CO2,</li>
	<li>v is the velocity of flow,</li>
	<li>D is the pipe diameter,</li>
	<li>&#956; is the dynamic viscosity of CO2.</li>
</ol>
The flow rate of CO2&#160;in this system is controlled by the pipe&#39;s cross-sectional area and the velocity of the gas. To maintain turbulent conditions, the flow velocity v is adjusted based on the desired Reynolds number, which allows the required pipe diameter to be calculated.
The Reynolds number equation can be rearranged to determine the pipe diameter for the required turbulent flow. The&#160;dynamic viscosity &#956; of CO2&#160;at 20&#176;C are obtained using physical property tables. The optimal pipe diameter can be calculated by substituting these values along with the target Reynolds number and flow velocity into the equation. This configuration ensures efficient CO2&#160;transport through the system, optimizing the liquid&#39;s carbonation conditions.

Consider a liquid that needs to be carbonated. Both the liquid and the carbon dioxide gas are cooled. The temperature is selected as 20 degrees Celsius to ...

General Characteristics of Pipe Flow II

When fluid enters a pipe, it first passes through the entrance region, where the velocity profile adjusts due to viscous effects. In this region, a boundary layer forms along the pipe walls and grows until it fully occupies the pipe&#39;s cross-section. Once the boundary layer merges, the flow becomes fully developed, with a steady velocity profile that remains consistent along the pipe&#39;s length.
The distance to reach a fully developed flow is called the entrance length and depends on the flow regime (laminar or turbulent) and the Reynolds number. Laminar flow typically has a shorter entrance length due to its smooth, gradual profile development, whereas turbulent flow requires a longer entrance length due to enhanced mixing and chaotic motion.
A pressure difference along the pipe&#39;s length drives fully developed flow in a constant-diameter pipe. In horizontal pipes, this pressure difference overcomes viscous forces, ensuring steady motion. The entrance region has a higher pressure gradient to account for acceleration as the velocity profile evolves. Once fully developed, the pressure gradient stabilizes, reflecting the energy needed to overcome viscosity.
Shear stress behaves differently in laminar and turbulent flows. In laminar flow, shear stress arises from molecular momentum transfer, resulting in smooth, orderly layers. Turbulent flow stems from momentum exchange between more significant, irregularly moving fluid particles, making it a macroscopic effect. These differences influence pressure distribution and flow characteristics.
For inclined pipes, gravity modifies the pressure gradient. Depending on the pipe&#39;s orientation, gravity can assist or resist flow, altering the force balance required for steady motion. The viscosity, pressure, and gravity interplay determine fluid behavior within the pipe.

As fluid enters a pipe, it first flows through an entrance region where the velocity profile adjusts, forming a boundary layer along the walls. This ...

Laminar Flow

Laminar flow represents a smooth, orderly fluid motion where particles move along parallel paths, resulting in minimal mixing between layers. Streamlined particle paths characterize this flow regime and occur under conditions where viscous forces dominate over inertial forces. The distinction between laminar, transitional, and turbulent flow is primarily determined by the Reynolds number, a dimensionless quantity calculated as:
<img alt="Equation 1" src="/files/ftp_upload/18076/18076_Equation 1.svg" style="text-align: center; height: 30px;" />
where &#961; is the fluid density, v is the flow velocity, D is the pipe diameter, and &#956; is the dynamic viscosity. A Reynolds number below approximately 2100 indicates laminar flow, while values above 4000 denote turbulent flow. Transitional flow occurs between these ranges as the fluid shifts from laminar to turbulent.
Velocity Profile and Entrance Length
In a pipe, fluid initially entering develops a velocity profile over a length known as the entrance length. During this phase, the flow transitions from a uniform profile at the inlet to a fully developed parabolic profile, where the velocity is highest at the center and gradually decreases toward the pipe walls due to viscous forces. The entrance length is a function of the Reynolds number and pipe characteristics, typically shorter in turbulent flow and longer in laminar flow. Once fully developed, the parabolic velocity profile remains stable unless disrupted by changes in pipe geometry, allowing consistent analysis of flow behavior along the pipe&#39;s length.
Energy Loss and Hagen-Poiseuille Law
Laminar flow is advantageous in minimizing energy loss, as the pressure drop along the pipe is dominated by viscous forces rather than turbulence. The relationship between pressure drop and flow rate in a fully developed laminar flow is defined by the Hagen-Poiseuille equation:
<img alt="Equation 2" src="/files/ftp_upload/18076/18076_Equation 2.svg" style="text-align: center; height: 30px;" />
where &#916;P is the pressure drop, &#956; is the dynamic viscosity, L is the pipe length, Q is the volumetric flow rate, and R is the pipe radius. This equation shows that the pressure drop in laminar flow is directly proportional to flow rate and fluid viscosity and inversely proportional to the fourth power of the pipe radius, highlighting the significant impact of pipe diameter on energy efficiency.
Applications of Laminar Flow
Laminar flow conditions are essential in applications that require smooth, predictable fluid motion with minimal mixing. Wastewater treatment systems, for example, benefit from laminar flow to avoid excessive energy loss and facilitate controlled fluid handling. In canal designs for irrigation, laminar flow promotes even distribution without turbulent disturbances, enhancing water delivery efficiency. Hydropower systems can also leverage laminar flow for more efficient energy conversion, particularly in sections where consistent, low-energy flow is needed.

Laminar flow refers to the smooth, orderly motion of fluid particles in which each particle follows parallel paths with minimal mixing between layers.
The ...

Laminar Flow: Problem Solving

Laminar flow occurs when a fluid moves smoothly in parallel layers with minimal mixing and turbulence. In fluid mechanics, ensuring laminar flow within a pipe is essential for precise control of flow characteristics, especially in engineering applications. The key factor in determining whether flow remains laminar is the Reynolds number, a dimensionless quantity that depends on the fluid&#39;s velocity, density, viscosity, and the pipe&#39;s diameter. A Reynolds number of 2100 or lower indicates laminar flow, while higher values lead to turbulence.
For air flowing through a pipe at standard atmospheric conditions and a given mass flow rate, the velocity of the fluid must be controlled to maintain laminar conditions. Since velocity depends on the flow rate and the cross-sectional area of the pipe, the pipe diameter plays a crucial role. By substituting the velocity expression into the Reynolds number formula and solving for the diameter, it is possible to determine the minimum required pipe size.
Using standard values for air density and viscosity at 300 Kelvin, the volumetric flow rate can be computed. This, in turn, allows for the calculation of the necessary pipe diameter. To ensure a Reynolds number of 2100 or lower, the minimum allowable diameter for the pipe is found to be approximately 0.46 meters. This calculation ensures that the flow remains smooth and predictable, avoiding turbulence that could disrupt performance.

Consider air flowing through the pipe at 300 Kelvin at standard atmospheric pressure, with a mass flowrate of 0.0163 kilograms per second. If the flow is ...

General Characteristics of Pipe Flow I

Pipe flow refers to the movement of fluids within fully enclosed conduits, typically cylindrical in shape, such as water pipes or hydraulic hoses. These conduits are designed to withstand high-pressure gradients that drive fluid movement, contrasting with open-channel flows, where gravity is the primary driving force. Rectangular conduits, like air conditioning and heating ducts, generally operate at lower pressures and are less suited for high-pressure applications.
The classification of fluid flow in pipes was pioneered by Osborne Reynolds through his foundational dye injection experiments. Reynolds identified three flow regimes &#8212; laminar, transitional, and turbulent &#8212; based on fluid velocity. At low velocities, the dye maintained a smooth streakline, indicating laminar flow, where fluid particles follow orderly, parallel paths. As velocity increased, the dye began to oscillate, marking the transitional regime, and at higher velocities, it dispersed unpredictably in the turbulent regime, where random, chaotic motion dominates.
These observations led to the development of the Reynolds number, a dimensionless quantity that indicates the balance of inertial and viscous forces in the flow. The Reynolds number is calculated based on fluid velocity, density, viscosity, and pipe diameter and serves as an indicator of flow type: laminar for values below 2100, turbulent for values above 4000, and transitional in the range between. In laminar flow, velocity remains steady over time, while turbulent flow exhibits unsteady, fluctuating velocities with random components perpendicular to the pipe axis. By calculating the Reynolds number, engineers can predict flow behavior in pipe systems, aiding in the design and optimization of pipelines, hydraulic systems, and fluid transport networks across various industries.

Pipe flow describes fluid movement through enclosed, fully-filled conduits, typically cylindrical pipes like water pipes or hydraulic hoses, engineered to ...

Design Example: Creating a Hydraulic Model of a Dam Spillway

Scaled hydraulic models of dam spillways provide a practical way to replicate and study the intricate flow dynamics of these structures. Often built to a 1:15 ratio, these models allow for observing critical water behavior, such as velocity distribution, flow patterns, and energy dissipation.
<img alt="Equation 1" src="/files/ftp_upload/18074/18074_Equation_1.svg" style="text-align: center; height: 38px;" />
Through this controlled setup, researchers can closely examine how water flows over the spillway, supporting a detailed analysis of its performance. A fundamental principle behind these models is using Froude number similarity, which preserves the proper balance between gravitational and inertial forces.
<img alt="Equation 2" src="/files/ftp_upload/18074/18074_Equation_2.svg" style="text-align: center; height: 38px;" />
This similarity is essential, as it ensures that free-surface flow characteristics, including wave formation, turbulence, and flow separation, behave as they would in the full-sized structure. By achieving this accurate representation, scaled models provide valuable insights into how water interacts with spillway features, helping to pinpoint areas where the design might benefit from adjustments to enhance safety or flow efficiency.
In the scaled model, adjustments are also applied to discharge rates, flow velocities, and timing to reflect real-world conditions without requiring extensive water volumes or prolonged test durations. The discharge rate, for example, is reduced by the scale factor to maintain manageable flow rates, enabling safe experimentation.
<img alt="Equation 3" src="/files/ftp_upload/18074/18074_Equation_3.svg" style="text-align: center; height: 36px;" />
Additionally, time is compressed so that events unfolding over an entire day in the full-scale structure can be examined in just a few hours within the model.
<img alt="Equation 4" src="/files/ftp_upload/18074/18074_Equation_4.svg" style="text-align: center; height: 48px;" />
This efficient scaling supports rapid testing and analysis of different flow conditions and potential structural responses. Overall, scaled modeling of spillways is crucial for assessing designs, identifying potential vulnerabilities, and confirming that the full-scale structure will perform reliably under various water flow conditions.

A hydraulic model of a dam spillway is often created by scaling down the design, typically to a 1:15 ratio, to replicate key flow characteristics such as ...

Dimensionless Groups in Fluid Mechanics

Dimensionless groups in fluid mechanics provide simplified ratios that help analyze fluid behavior without relying on specific units. The Reynolds number (Re), which represents the ratio of inertial to viscous forces, distinguishes between laminar and turbulent flows, making it essential in the design of pipelines and aerodynamic surfaces. The Froude number (Fr), the ratio of inertial to gravitational forces, is particularly useful in predicting wave formation and hydraulic jumps in open-channel flows and coastal engineering.
The Euler number (Eu) indicates the ratio of pressure forces to inertial forces and is used to calculate pressure drops in systems such as pipelines and across valves. The Cauchy number (Ca), representing the ratio of inertial to elastic forces, is critical in compressible fluid flow analysis and the study of shockwaves. The Mach number (Ma) compares the flow speed to the speed of sound, making it crucial for studies in supersonic flight and aeroelasticity.
The Strouhal number (St), which compares oscillatory frequency to inertial forces, finds applications in resonance studies, particularly vortex shedding around structures like bridge piers. Lastly, the Weber number (We), showing the ratio of inertial to surface tension forces, is vital in analyzing droplet formation, spray dynamics, inkjet printing, and fuel atomization processes. These groups enable the comparison of fluid behaviors across different scales and systems, supporting a broad range of fluid dynamics applications.

The dimensionless groups in fluid mechanics are ratios of physical quantities used to describe and analyze fluid behavior independently of units of ...

Correlation of Experimental Data

Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
For example, a spherical particle moving through a viscous fluid experiences drag. Dimensional analysis shows that the drag force depends on the particle&#39;s diameter, velocity, and the fluid&#39;s viscosity. These variables are grouped into dimensionless terms to form a relationship, but experimental data is needed to determine the exact nature of this relationship.
Researchers measure drag force while varying velocity to correlate the dimensionless groups, such as the Reynolds number. This correlation allows the relationship between drag and velocity to be expressed as a constant. Multiple tests verify this constant, confirming the dependency of drag on velocity.
If drag depends only on diameter, velocity, and viscosity, the relationship should be universal for any spherical particle in any fluid under consistent conditions. Multiple tests with different particles and fluids verify this universality, reducing the need for further experiments, as the derived constant remains valid under the specified conditions.

A key use of dimensional analysis is to efficiently manage, interpret, and correlate experimental data.
Dimensional analysis alone cannot fully solve a ...

Typical Model Studies

Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
<img alt="Equation 1" src="/files/ftp_upload/18073/18073_Equation_1.svg" style="text-align: center; height: 48px;" />
Practical constraints, however, can introduce geometric distortions, impacting how accurately these models represent the real system. Adjustments, such as increased model roughness, help bridge this gap, aligning flow behaviors between the model and the actual river environment.
For hydraulic structures like spillways, the focus shifts to achieving high Reynolds numbers to maintain dynamic similarity. This approach emphasizes balancing inertial and gravitational forces to replicate the prototype&#39;s behavior closely.
<img alt="Equation 2" src="/files/ftp_upload/18073/18073_Equation_2.svg" style="text-align: center; height: 45px;" />
Larger models are generally preferred, as they capture critical force interactions more effectively in full-scale designs. Although exact similarity across all scales may not be feasible due to physical and financial limitations, these larger models still allow close approximations of essential conditions.
Verification tests and prototype comparisons are key for refining model accuracy. Adjusting variables and calibrating outcomes ensure that the model reliably mirrors actual fluid flow behaviors. This predictive capability supports planning and design adaptations, making these models indispensable tools for assessing performance before moving to full-scale implementation.

Typical model studies analyze scaled-down systems to predict fluid behavior, focusing on interactions with free surfaces and flow around immersed bodies.
...

Modeling and Similitude

Scaled modeling is a fundamental technique in engineering, enabling the study of large and complex systems by creating smaller, manageable replicas that recreate critical characteristics of the original. In hydrology and civil infrastructure, for example, scaled models of dams help analyze water flow, turbulence, and pressure. This method allows for accurate predictions of real-world behavior within a controlled environment, significantly reducing the cost and time involved in full-scale testing, which is often impractical. These models provide valuable insights into system performance under various conditions by ensuring that essential design elements match.
For the model to yield valid insights, it must maintain similitude, meaning the replica accurately reflects the prototype&#39;s behavior through geometric, kinematic, and dynamic similarity. Geometric similarity ensures the scaled model retains the same proportional dimensions as the actual structure, while kinematic similarity requires that motion characteristics match between model and prototype. Dynamic similarity further demands that forces acting within the model accurately replicate those in the real system, accounting for any scaling effects.
In fluid dynamics, the Froude number is a key dimensionless parameter that matches scaled models to their full-sized counterparts, especially for applications like ship design. The Froude number is defined as:
<img alt="Equation 1" src="/files/ftp_upload/18072/18072_Equation_1.svg" style="text-align: center; height: 60px;" />
where V represents velocity, g is the gravitational acceleration, and L is a characteristic length, often the ship&#39;s length. By equating the Froude numbers for both the model and the full-scale ships, the gravitational and inertial forces align, allowing the scaled model to accurately predict behaviors such as maneuverability, stability, and wave formation.

A scaled model of a dam is tested in a controlled environment to observe how it will manage water flow, turbulence, and pressure.
This setup predicts the ...

Determination of Pi Terms

The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions &#8212; mass, length, and time &#8212; provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the number of Pi terms equals the total variables minus the number of fundamental dimensions. In this example, five variables and three dimensions result in two Pi terms. Air density, velocity, and wing area are selected as repeating variables, as they independently cover all dimensions. These repeating variables combine with lift force and viscosity to form dimensionless Pi terms.
<img alt="Equation 1" src="/files/ftp_upload/18069/18069_Equation_1.svg" style="text-align: center; height: 100px;" />
The first Pi term expresses the efficiency of lift generation, representing how lift depends on density, velocity, and area. The second Pi term accounts for the relative impact of viscosity, revealing how viscous forces interact with the wing in relation to inertia.
Each Pi term is verified as dimensionless, allowing complex aerodynamic behavior to be represented in simplified expressions. This approach enables the prediction and optimization of lift under various conditions, capturing the essential dynamics without needing to examine each factor individually.

To determine Pi terms using the Buckingham Pi theorem, consider the example of analyzing lift force on an airplane wing.
Relevant variables here include ...

Dimensional Analysis

Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
In fluid mechanics, dimensional analysis is used to understand the behavior of fluids in systems such as pipelines, channels, and other structures common in civil engineering. Consider the example of fluid flow through a long, smooth-walled, horizontal circular pipe, which is typical in the design of water distribution systems or sewage networks. The pressure drop per unit length along the pipe depends on variables such as pipe diameter, fluid density, fluid velocity, and viscosity. According to dimensional analysis, the pressure drop can be expressed as a function of these variables:
<img alt="Equation 1" src="/files/ftp_upload/18067/18067_Equation_1.svg" style="text-align: center; height: 12px;" />
This formulation involves five independent variables, which can complicate the analysis. To simplify this, we can use dimensional analysis to reduce the number of variables into two dimensionless groups, thus streamlining the problem. Two important dimensionless groups that emerge from this analysis are the Reynolds number, which characterizes the flow regime, and the friction factor, which correlates with the pressure drop.
<img alt="Equation 2" src="/files/ftp_upload/18067/18067_Equation_2.svg" style="text-align: center; height: 30px;" />
Engineers use the dimensionless groups to generate a universal curve that applies to any smooth-walled pipe and incompressible Newtonian fluid. This curve would allow them to predict the pressure drop for different pipe sizes and fluids without performing numerous experiments for each possible combination.
<img alt="Figure 1" src="/files/ftp_upload/18067/18067_PT_Figure.svg" style="text-align: center; height: 250px;" />
This approach minimizes the time and cost required for experimentation and allows for more straightforward design decisions. Whether designing water distribution networks or analyzing pressure losses in irrigation systems, dimensional analysis is critical for any civil engineer.

Dimensional analysis helps simplify fluid flow problems by using dimensionless groups, reducing complex variables into simpler terms that are easier to ...

The Buckingham Pi Theorem

The Buckingham Pi theorem provides a structured method to simplify fluid dynamics problems by reducing complex systems of variables to dimensionless terms.
<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;" />
In analyzing flow through a pipe, variables that define the system include the pipe&#39;s diameter, the fluid&#39;s velocity, its density, and viscosity. These can be combined into a single dimensionless term, the Reynolds number (Re), which captures their essential relationship.
<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;" />
The first step in applying the Buckingham Pi theorem is identifying the fundamental dimensions of each variable. In fluid mechanics, these are typically mass, length, and time. Here, the diameter represents length, velocity combines length and time, density encompasses mass and volume, and viscosity incorporates mass, length, and time.
<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;" />
The theorem suggests that the number of independent dimensionless groups, known as Pi terms, is the difference in the number of variables k and the number of fundamental dimensions r. Only one dimensionless term is needed with four variables and three dimensions, simplifying analysis.
<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;" />
This single term, the Reynolds number, compares inertial forces to viscous forces in the fluid. Establishing this ratio reveals whether the flow is likely to be smooth and orderly, known as laminar flow, or chaotic and irregular, known as turbulent flow. Lower values indicate laminar flow, where viscous forces dominate, and the fluid moves in parallel layers. Higher values reflect turbulent flow, where inertial forces lead to chaotic motion. The Buckingham Pi theorem&#39;s application in this context reduces complex interactions to a single term, making it easier to generalize flow behavior under varying conditions.

Consider the flow through a pipe, where key variables include the pipe&#39;s diameter D, fluid velocity V, density &#961;, and viscosity &#956;.
To ...

Navier&#8211;Stokes Equations

For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting tangential stress to velocity differences between layers. Normal and shear stresses define the internal forces within a fluid that influence its movement.
The Navier-Stokes equations are derived by incorporating these stress relationships into differential equations of motion.
<img alt="Equation 1" src="/files/ftp_upload/18062/18062_Equation_1.svg" style="text-align: center; height: 106px;" />
These equations balance inertial, pressure, and gravitational forces in a viscous fluid and are fundamental to predicting fluid behavior under various conditions. The inertia term in the Navier-Stokes equations accounts for fluid acceleration, representing the fluid&#39;s resistance to changes in velocity. This term is critical in capturing the momentum conservation of a moving fluid, as it demonstrates how fluids resist sudden shifts in speed or direction.
The pressure gradient term within the Navier-Stokes equations drives fluid movement by creating a net force from areas of high pressure to low pressure. Meanwhile, the viscous terms represent internal friction arising from molecular interactions within the fluid. Depending on the velocity gradients present, these viscous forces act to slow down the fluid movement.
Each directional Navier-Stokes equation captures internal forces like viscous stress and external forces like gravity. These equations allow for predicting fluid motion in various conditions, from simple to complex flows. For steady or laminar flow scenarios, where flow is smooth and orderly, the equations become more manageable, enabling more straightforward analysis. This simplification is useful in controlled scenarios like boundary layer flow, where flow near a surface has a structured layer, and in Couette flow, describing fluid between two parallel surfaces moving at different velocities.

In fluid motion under viscous conditions, shear stress is directly proportional to fluid deformation. In incompressible Newtonian fluids, this stress ...

Plane Potential Flows

Plane potential flows simplify fluid motion by assuming the fluid to be irrotational and incompressible. These characteristics allow these flows to be described by a velocity potential function, &#981;, representing the flow speed in a given direction, and a stream function, &#968;, that visualizes the flow path, both governed by Laplace&#39;s equation. These parameters help in estimating flow patterns, velocity distributions, and pressure fields around various hydraulic structures.
Uniform Flow
Uniform flow represents the simplest form of potential flow, with fluid moving in a straight line at a constant velocity. Streamlines are straight, parallel, and evenly spaced, clearly visualizing flow across channels or pipes with constant velocity. This flow type is essential in designing canals, pipelines and drainage systems, where the assumption of constant velocity aids in flow rate estimation and hydraulic design.
Source and Sink Flows
Source and sink flows describe radially symmetric flows originating from or converging to a point. In a source flow, fluid moves radially outward from a central point, while in a sink flow, it converges inward. These flows are modeled using a flow rate parameter to represent the strength of the source or sink. This model is widely used in groundwater hydrology, where source flows represent groundwater recharge, such as that from an injection well, and sink flows represent groundwater extraction in a pumping well.&#160; represent groundwater extraction from a pumping well., and sink flows simulate groundwater recharge .
Vortex Flow
Vortex flow involves fluid circulating a central point and appears in two forms: free and forced. In a free vortex, particles move in circular paths around a center point without external force, creating a whirlpool effect often observed in natural bodies of water. This model is used to design spillways and drainage systems that manage spiraling water flow to prevent clogging or turbulence. In a forced vortex, an external force drives the fluid&#39;s rotation, creating a scenario where the fluid rotates as a rigid body. Forced vortices are used in sedimentation tanks and certain hydraulic devices where uniform rotation aids in sediment separation and fluid mixing.
Flow Nets
In flow nets, streamlines represent the fluid flow path, while equipotential lines mark points of equal hydraulic potential. These lines intersect perpendicularly, creating a grid pattern that aids in identifying pressure distributions and potential seepage paths around structures. Flow nets are constructed to assess seepage in soil or porous media. Seepage rates are estimated by carefully analyzing the spacing and orientation of these lines and identifying potential areas of erosion or structural weakness.

Plane potential flows simplify fluid motion by assuming irrotational and incompressible flow, governed by a velocity potential function and a stream ...

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
Considering two horizontal plates aligned along the x-axis and spaced a distance &#39;2h&#39; apart, with the y-axis perpendicular to the plates, fluid motion is assumed to be parallel. Here, the continuity equation indicates no change in velocity along the x-axis, and the Navier-Stokes equations reduce as follows:
<img alt="Equation 1" src="/files/ftp_upload/18063/18063_Equation_1.svg" style="text-align: center; height: 30px;" />
The equations in the y and z directions show no pressure gradient along these axes, indicating that pressure varies hydrostatically in the y-direction, allowing integration to yield:
<img alt="Equation 2" src="/files/ftp_upload/18063/18063_Equation_2.svg" style="text-align: center; height: 12px;" />
For parallel plates with a pressure gradient along the x-axis, the equation simplifies:
<img alt="Equation 3" src="/files/ftp_upload/18063/18063_Equation_3.svg" style="text-align: center; height: 30px;" />
By integrating twice and applying boundary conditions (u = 0 at y = +h and y = -h), we obtain:
<img alt="Equation 4" src="/files/ftp_upload/18063/18063_Equation_4.svg" style="text-align: center; height: 30px;" />
This expression reveals that the velocity profile across the gap between the plates is parabolic, with maximum velocity at y = 0, midway between the plates. The volumetric flow rate through a unit width of the plates is obtained by integrating the velocity profile across the gap.
Understanding laminar flow between parallel plates has practical applications in groundwater flow, wastewater treatment, and lubrication. For instance, this model can approximate flow in soil layers, where water moves between soil particles like flow between plates. Knowing the flow rate and velocity distribution helps engineers design efficient drainage systems and predict groundwater movement.

Steady, laminar flow between two parallel plates describes fluid movement in a narrow, confined channel, often used to model simple fluid systems.
For ...

Aerodynamic Performance of a Model Aircraft: The DC-6B

Source: Jose Roberto Moreto and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
The low-speed wind tunnel is a valuable tool to study aircraft aerodynamic characteristics and evaluate aircraft performance and stability. Using a scale model of a DC-6B aircraft that has a removable tail and a 6-component external aerodynamic force balance, we can measure the lift coefficient (CL), drag coefficient (CD), pitching moment coefficient (CM), and yaw moment coefficient (CN) of the model airplane with and without its tail and evaluate the effect of the tail on aerodynamic efficiency, longitudinal stability and directional stability.
In this demonstration, airplane aerodynamic characteristics and flight performance and stability are analyzed using the aerodynamic force balance measurement method. This method is widely used in aerospace industries and research labs for aircraft and rocket development. Here, a model DC-6B airplane is analyzed at different flow conditions and configurations, and its behavior is analyzed when it is subjected to sudden changes.

Forces at a Non-zero Windspeed in the DC-6B Model Aircraft

Source: Jose Roberto Moreto and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
The low-speed wind tunnel is ...

Propeller Characterization: Variations in Pitch, Diameter, and Blade Number on Performance

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
A propeller is a twisted airfoil, where the angle of the chord changes with respect to the location, along the radial station, as shown in Figure 1. Propellers are widely used in aircraft and watercraft propulsion systems thereby necessitating detailed characterizations of propellers to design high performance vehicles.
<img src="/files/ftp_upload/10460/10460fig1.jpg" />
Figure 1. Chord, thickness, and pitch at a radial station.
One of the defining characteristics of a propeller is the pitch/twist. The pitch of the propeller, generally given in units of length, is the theoretical distance the propeller will travel through the air in one single revolution. However, due to the drag force on the aircraft and the propeller, the propeller never travels its theoretical distance. The actual distance travelled is referred to as the effective pitch of the propeller, and the difference between the theoretical or geometric pitch and the effective pitch is referred to as propeller slip, as illustrated in Figure 2.
<img src="/files/ftp_upload/10460/10460fig2.jpg" />
Figure 2. Representation of pitch and slip.
In this demonstration, seven propellers are characterized using a propeller test rig in a subsonic wind tunnel. This is followed by a detailed parametric study to analyze the effects of variations in pitch, diameter, and number of blades on propeller performance.

Propeller Characterization: Pitch, Diameter and Blade Number

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
A propeller is a twisted airfoil, where the ...

Airfoil Behavior: Pressure Distribution over a Clark Y-14 Wing

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
An airfoil is a 2-dimensional wing section that represents critical wing performance characteristics. The pressure distribution and lift coefficient are important parameters that characterize the behavior of airfoils. The pressure distribution is directly related to the lift generated by airfoils. A Clark Y-14 airfoil, which is used in this demonstration, has a thickness of 14% and is flat on the lower surface from 30% of chord length to the back.
Here we will demonstrate how the pressure distribution around an airfoil is measured using a wind tunnel. A Clark Y-14 airfoil model with 19 pressure ports is used to collect pressure data, which is used to estimate the lift coefficient.

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
An ...

Clark Y-14 Wing Performance: Deployment of High-lift Devices (Flaps and Slats)

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
A wing is the major lift-generating apparatus in an airplane. Wing performance can be further enhanced by deploying high-lift devices, such as flaps (at the trailing edge) and slats (at the leading edge) during takeoff or landing.
In this experiment, a wind tunnel is utilized to generate certain airspeeds, and a Clark Y-14 wing with a flap and slat is used to collect and calculate data, such as the lift, drag and pitching moment coefficient. A Clark Y-14 airfoil is shown in Figure 1 and has a thickness of 14% and is flat on the lower surface from 30% of the chord to the back. Here, wind tunnel testing is used to demonstrate how the aerodynamic performance of a Clark Y-14 wing is affected by high-lift devices, such as flaps and slats.
<img src="/files/ftp_upload/10454/10454fig1.jpg" />
Figure 1. Clark Y-14 airfoil profile.

Clark Y-14 Wing Performance: Deployment of High-lift Devices

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
A wing ...

Turbulence Sphere Method: Evaluating Wind Tunnel Flow Quality

Source: Jose Roberto Moreto and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
Wind tunnel tests are useful in the design of vehicles and structures that are subjected to airflow during their use. Wind tunnel data are generated by applying a controlled air flow to a model of the object being studied. The test model usually has a similar geometry but is a smaller scale compared to the full-sized object. To ensure accurate and useful data is collected during low speed wind tunnel tests, there must be a dynamic similarity of the Reynolds number between the tunnel flow field over the testing model and the actual flow field over the full-sized object.
In this demonstration, wind tunnel flow over a smooth sphere with well-defined flow characteristics will be analyzed. Because the sphere has well-defined flow characteristics, the turbulence factor for the wind tunnel, which correlates the effective Reynolds number to the test Reynolds number, can be determined, as well as the free-stream turbulence intensity of the wind tunnel.

Turbulence Measurements at Varying Air Speeds

Source: Jose Roberto Moreto and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
Wind tunnel tests are useful ...

Cross Cylindrical Flow: Measuring Pressure Distribution and Estimating Drag Coefficients

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
The pressure distributions and drag estimations for cross cylindrical flow have been investigated for centuries. By ideal inviscid potential flow theory, the pressure distribution around a cylinder is vertically symmetric. The pressure distribution upstream and downstream of the cylinder is also symmetric, which results in a zero-net drag force. However, experimental results yield very different flow patterns, pressure distributions and drag coefficients. This is because the ideal inviscid potential theory assumes irrotational flow, meaning viscosity is not considered or taken into account when determining the flow pattern. This differs significantly from reality.
In this demonstration, a wind tunnel is utilized to generate a specified airspeed, and a cylinder with 24 ports of pressure is used to collect pressure distribution data. This demonstration illustrates how the pressure of a real fluid flowing around a circular cylinder differs from predicted results based on the potential flow of an idealized fluid. The drag coefficient will also be estimated and compared to the predicted value.

Cross Cylindrical Flow: Pressure Distribution &amp; Drag Coefficients

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
The ...

Nozzle Analysis: Variations in Mach Number and Pressure Along a Converging and a Converging-diverging Nozzle

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
A nozzle is a device that is commonly used to accelerate or decelerate flow by virtue of its varying cross-section. Nozzles are widely used in aerospace propulsion systems. In rockets, propellant that is ejected from the chamber is accelerated through a nozzle to create a reaction force that propels the system. In jet engines, a nozzle is used to transform energy from a high-pressure source into kinetic energy of the exhaust to produce thrust. The isentropic model along the nozzle is sufficient for a first-order analysis as the flow in a nozzle is very rapid (and thus adiabatic to a first approximation) with very little frictional loses (because the flow is nearly one-dimensional with a favorable pressure gradient, except if shock waves form and nozzles are relatively short).
In this experiment, two types of nozzles are mounted on a nozzle test rig, and a pressure flow is created using a compressed air source. The nozzles are run for different back-pressure settings to analyze the internal flow in the nozzles under varying flow conditions, identify the various flow regimes, and compare the data to theoretical predictions.

Nozzle Analysis: Mach Number and Converging-diverging Nozzle

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
A nozzle is a device that is commonly used to ...

Schlieren Imaging: A Technique to Visualize Supersonic Flow Features

Source: Jose Roberto Moreto, Jaime Dorado, and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
Military jet fighters and projectiles can fly at incredible speeds that exceed the speed of sound, which means they are traveling at a supersonic speed. The speed of sound is the speed at which a sound wave propagates through a medium, which is 343 m/s. Mach numbers are used to gauge the flight speed of an object relative to the speed of sound.
An object traveling at the speed of sound would have a Mach number of 1.0, whereas an object traveling faster than the speed of sound has a Mach number greater than 1.0. The compressibility effects of air must be accounted for when traveling at such speeds. A flow is considered compressible when the Mach number is greater than 0.3. In this demonstration, Mach 2.0 supersonic flow over a cone will be analyzed by visualizing the formation of shock waves and compression waves in compressible flow using a Schlieren system.

Schlieren Imaging: Visualization of Supersonic Flow Features

Source: Jose Roberto Moreto, Jaime Dorado, and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, CA
Military jet ...

Flow Visualization in a Water Tunnel: Observing the Leading-edge Vortex Over a Delta Wing

Source: Jose Roberto Moreto, Gustaaf Jacobs and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, California
The delta wing, shown in Figure 1D, is a popular design in high-speed airplanes due to its superb performance in transonic and supersonic flight regimes. This type of wing has a small aspect ratio and high sweep angle, which reduces drag at high subsonic, transonic and supersonic flight regimes. The aspect ratio is defined as the wing span divided by the average chord <img align="middle" src="/files/ftp_upload/10459/10459eq1a.jpg" />.
An important advantage of the delta wing is its high stall angle. The stall of a delta wing is delayed compared to the stall of a high aspect ratio wing. This is because the lift of a delta wing is enhanced by the leading-edge vortex over the wing.
An effective way to observe this vortex flow phenomenon and study the vortex breakdown in a delta wing is by visualizing the flow in a water tunnel. By injecting dye in the flow surrounding a model from dye ports on the leading-edge, the vortex development and breakdown can be observed and its position measured. The data can also be used to estimate the stall angle.
<img src="/files/ftp_upload/10459/10459fig1.jpg" />
Figure 1. Typical wing planform shapes: A) Rectangular, with constant chord along span, B) elliptical, C) tapered, with variable chord along the span, and D) delta wing, an aft-swept wing with zero taper ratio.

Flow in a Water Tunnel: Leading-edge Vortex over a Delta Wing

Source: Jose Roberto Moreto, Gustaaf Jacobs and Xiaofeng Liu, Department of Aerospace Engineering, San Diego State University, San Diego, California
The ...

Surface Dye Flow Visualization: A Qualitative Method to Observe Streakline Patterns in Supersonic Flow

Flow visualization around or on a body is an important tool in aerodynamics research. It provides a method to qualitatively and quantitatively study flow structure, and it also helps researchers theorize and verify fluid flow behavior. Flow visualization can be divided into two categories: off-the-surface visualization and surface flow visualization. Off-the-surface flow visualization techniques involve determining the flow characteristics around the body of interest. They include but are not restricted to particle image velocimetry (PIV), Schlieren imaging, and smoke flow visualization. These techniques can provide qualitative as well as quantitative data on the flow around a body. However, these techniques are generally expensive and difficult to set-up. Surface flow visualization techniques, on the other hand, involve coating the body of interest with a dye to study flow on the surface. These techniques, which are more invasive in practice, include dye flow visualization and, more recently, use pressure-sensitive paint, which gives a detailed image of the flow on the body&#39;s surface. This allows researchers to visualize different flow features, including laminar bubbles, boundary layer transitions, and flow separation. Dye flow visualization, the technique of interest in the current experiment, provides a qualitative picture of the surface flow and is one of the simplest and most cost-effective surface flow visualization methods, specifically for visualizing gaseous flows on a body.
In this experiment, the surface flow behavior on six bodies are studied in supersonic flow. The streakline patterns are obtained using the dye flow visualization technique, and the flow paths, degree of flow attachment and separation, and location and type of shocks are identified and studied from the flow images.

Surface Dye Flow: Streakline Patterns in Supersonic Flow

Flow visualization around or on a body is an important tool in aerodynamics research. It provides a method to qualitatively and quantitatively study flow ...

Pitot-static Tube: A Device to Measure Air Flow Speed

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
A Pitot-static tube is widely used for measuring unknown speeds in air flow, for example, it is used to measure airplane airspeed. By Bernoulli's principle, airspeed is directly related to variations in pressure. Therefore, the Pitot-static tube senses the stagnation pressure and static pressure. It is connected to a manometer or pressure transducer to obtain pressure readings, which allows airspeed prediction.
In this experiment, a wind tunnel is utilized to generate certain airspeeds, which is compared against Pitot-static tube predictions. The sensitivity of the Pitot-static tube due to misalignment with respect to flow direction is also investigated. This experiment will demonstrate how air flow speed is measured using a Pitot-static tube. The goal will be to predict the air flow speed based on the pressure measurements obtained.

Source: David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University (SNHU), Manchester, New Hampshire
A ...

Constant Temperature Anemometry: A Tool to Study Turbulent Boundary Layer Flow

Source: Xiaofeng Liu, Jose Roberto Moreto, and Jaime Dorado, Department of Aerospace Engineering, San Diego State University, San Diego, California
A boundary layer is a thin flow region immediately adjacent to the surface of a solid body immersed in flow field. In this region, viscous effects, such as the viscous shear stress, dominate, and the flow is retarded due to the influence of friction between the fluid and the solid surface. Outside of the boundary layer, the flow is inviscid, i.e., there is no dissipative effects due to friction, thermal conduction or mass diffusion.
The boundary layer concept was introduced by Ludwig Prandtl in 1904, which enables significant simplification to the Navier-Stokes (NS) equation for the treatment of flow over a solid body. Inside the boundary layer, the NS equation is reduced to the boundary layer equation, while outside of the boundary layer, the flow can be described by the Euler equation, which is a simplified version of the NS equation.
<img src="/files/ftp_upload/10455/10455fig1a.jpg" />
Figure 1. Boundary layer development over a flat plate.
The simplest case for boundary layer development occurs on a flat plate at zero angle of incidence. When considering boundary layer development on a flat plate, the velocity outside of the boundary layer is constant so that the pressure gradient along the wall is considered to be zero.
The boundary layer, which naturally develops on a solid body surface, typically undergoes the following stages: first, the laminar boundary layer state; second, the transition state, and third, the turbulent boundary layer state. Each state has its own law(s) describing the flow structure of the boundary layer.
The research into the development and structure of boundary layer is of great importance for both theoretical study and practical applications. For example, boundary layer theory is the basis for calculating the skin friction drag on ships, aircraft and the blades of turbomachines. Skin friction drag is created on the body surface within the boundary layer and is due to the viscous shear stress exerted on the surface through fluid particles in direct contact with it. The skin friction is proportional to the fluid viscosity and the local velocity gradient on the surface in the surface normal direction. Skin friction drag is present on the entire surface, thus it becomes significant over large areas, such as an airplane wing. Additionally, turbulent fluid flow creates more skin friction drag. The macro-turbulent fluid motion enhances the momentum transfer within the boundary layer by bringing fluid particles with high momentum down to the surface.
This demonstration focuses on the turbulent boundary layer over a flat plate, in which the flow is irregular, such as in mixing or eddying, and the fluctuations are superimposed on the mean flow. Thus, the velocity at any point in a turbulent boundary layer is a function of time. In this demonstration, constant temperature hot-wire anemometry, or CTA, will be used to conduct a boundary layer survey. Then, the Clauser chart method will be used to calculate the skin friction coefficient in a turbulent boundary layer.

Constant Temperature Anemometry: Turbulent Boundary Layer Flow Study

Source: Xiaofeng Liu, Jose Roberto Moreto, and Jaime Dorado, Department of Aerospace Engineering, San Diego State University, San Diego, California
A ...

Pressure Transducer: Calibration Using a Pitot-static Tube

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
Fluid pressure is an important flow characteristic that is required to determine the aerodynamics of a system. One of the oldest and still existing pressure measurement systems is the manometer due to its accuracy and simplicity of operation. The manometer is generally a U-shaped glass tube that is partially filled with liquid, as shown in Figure 1. The U-tube manometer requires no calibration because it does not have any moving parts, and its measurements are functions of gravity and the liquid&#39;s density. Therefore, the manometer is a simple and accurate measurement system.
<img src="/files/ftp_upload/10467/10467fig1.jpg" /> 
Figure 1. Schematic of a U-tube manometer.
Real-time pressure measurements are obtained in aircraft by connecting the stagnation and static pressure ports of a pitot-static probe, a device that is commonly used to measure fluid flow pressure, to the ports of a pressure measurement device. This allows pilots to obtain existing flight conditions and to warn them if any changes to the flight conditions occur. While manometers provide very accurate pressure readings, they are inherently bulky. A more realistic solution is needed to measure aircraft pressures, as one of the primary design objectives is to keep the overall aircraft weight as low as possible. Today, electromechanical pressure transducers, which convert the applied pressure to an electric signal, are widely used for pressure sensing applications on aircraft because they are small, lightweight, and can be placed almost anywhere in the airframe. The above characteristics not only help reduce weight but also reduce the amount of tubing required to connect the pitot-static probe to the transducer, thereby decreasing the data response time. Additionally, in experimental aircraft flight testing, miniature pressure transducers come in handy as they allow researchers to maximize pressure data collection without significantly adding to the weight of the aircraft. While different types of pressure transducers with varying measurement techniques exist, one of the more common types of transducer is the capacitive pressure transducer. As transducers are capable of only sending signals in terms of voltage and current, calibration of the transducer is required to relate the strength of a particular signal to the pressure that causes the transducer to generate the signal. The final curve fit that relates the transducer current or voltage to a physical measurement, in our case pressure, is commonly referred to as the transducer calibration curve.
In this experiment, a pitot-static probe is placed in a subsonic wind tunnel with the stagnation and static pressure ports connected to the total and static ports of both the U-tube manometer and the pressure transducer. The wind tunnel is then run at different dynamic pressure settings, and the corresponding pressure reading from the U-tube manometer, and current readings produced by the transducer are recorded. This data is then used to generate calibration curves for the pressure transducer.

Calibration of Pressure Transducer Using a Pitot-static Tube

Source: Shreyas Narsipur, Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC
Fluid pressure is an important flow ...

Real-time Flight Control: Embedded Sensor Calibration and Data Acquisition

Source: Ella M. Atkins, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI
Overview
Autopilot allows aircraft to be stabilized using data collected from onboard sensors that measure the aircraft&#8217;s orientation, angular velocity, and airspeed. These quantities can be adjusted by the autopilot so that the aircraft automatically follows a flight plan from launch (takeoff) through recovery (landing). Similar sensor data is collected to control all types of aircraft, from large fixed-wing commercial transport aircraft to small-scale multiple-rotor helicopters, such as the quadcopter with four thruster units.
With inertial position and velocity captured by a sensor such as the Global Positioning System (GPS), the autopilot real-time flight control system enables a multicopter or fixed-wing aircraft to stabilize its attitude and airspeed to follow a prescribed trajectory. Sensor integration, calibration, data acquisition, and signal filtering are prerequisites for experiments in flight control.
Here we describe a sensor suite that provides the necessary data for flight control. Signal interfaces and data acquisition on two different embedded computer platforms are described, and sensor calibration is summarized. Single-channel moving average and median filters are applied to each data channel to reduce high-frequency signal noise and eliminate outliers.
In this experiment, data acquisition and sensor calibration for real-time flight control is demonstrated. Several published papers have described the principles of sensor data collection and control, and they have recently focused on sensors for small unmanned aerial vehicles (UAVs) [1-3].

Real-time Flight Control: Embedded Sensor Calibration

Source: Ella M. Atkins, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI
Overview
Autopilot allows aircraft to be stabilized ...

Multicopter Aerodynamics: Characterizing Thrust on a Hexacopter

Source: Prashin Sharma and Ella M. Atkins, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI
Multicopters are becoming popular for a variety of hobby and commercial applications. They are commonly available as quadcopter (four thrusters), hexacopter (six thrusters), and octocopter (eight thrusters) configurations. Here, we describe an experimental process to characterize the multicopter performance. A modular small hexacopter platform providing propulsion unit redundancy is tested. The individual static motor thrust is determined using a dynamometer and varying propeller and input commands. This static thrust is then represented as a function of motor RPM, where the RPM is determined from motor power and control input. The hexacopter is then mounted on a load cell test stand in a 5&#8217; x 7&#8217; low-speed recirculating wind tunnel, and its aerodynamic lift and drag force components were characterized during flight at varying motor signals, free-stream flow speed, and angle of attack.

A hexacopter was selected for this study because of its resilience to motor (propulsion unit) failure, as reported in Clothier1. Along with redundancy in the propulsion system, the selection of high-reliability components is also required for safe flight, particularly for missions over-populated regions. In Ampatis2, the authors discuss the optimal selection of multicopter parts, such as motors, blades, batteries, and electronic speed controllers. Similar research has also been reported in Bershadsky3, which focuses on the proper selection of a propeller system to satisfy mission requirements. Along with redundancy and reliability of components, understanding vehicle performance is also essential to assure flight envelope limits are respected and to select the most efficient design.

Characterizing Thrust on Hexacopter: Dynamometer Experiment

Source: Prashin Sharma and Ella M. Atkins, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI
Multicopters are becoming popular ...

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

@media screen and (min-width: 320px){.css-u03n47{font-size:var(--chakra-fontSizes-xs);}}Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)

Science Education (Engineering)