The present research work was financially sponsored by the Ministry of Science and Technology of Taiwan under the grant NSC 94-2611-E-022-001, NSC 95-2221-E-022-018, NSC 96-2221-E-022-015MY3 and NSC 97-2221-E-022-013-MY3.

NOTE: The details of rotating test facilities, data acquisition, data processing and the heat transfer test module emulating an internal cooling channel of a gas turbine rotor blade are in our previous works29,30,31,32.
1. Preparation of Heat Transfer Tests
<ol>
	<li>Formulate the experimental conditions in terms of Re, Ro and Bu from the targeted operation conditions of a gas turbine rotor blade.</li>
	<li>Determine the N, P, d, R, and Tw - Tb needed for acquiring the tested Re, Ro and Bu using equations (14) and (15).</li>
	<li>Re-define the targeting Re, Ro and Bu if N, P, d, R, and Tw - Tb exceeds the limit of the experimental facilities.</li>
	<li>Design and construct the scaled heat transfer test module emulating a practical internal coolant channel in a gas turbine rotor blade2.</li>
</ol>
2. Determination of Thermal Emissivity Coefficient for Infrared Thermography System
<ol>
	<li>Install the calibrated thermocouple on the back side of the scanned stainless-steel heating foil.</li>
	<li>Spray a thin layer of black paint on the stainless-steel heating foil scanned by the infrared camera.</li>
	<li>Create symmetrical flow fields on two sides of the stainless-steel heating foil by placing a vertical thin stainless-steel foil in a space with the free convective flows over the two sides of the vertical heating foil.</li>
	<li>Feed electrical heating power through the heating foil and measure temperatures simultaneously by thermocouple and infrared thermography system from the computer display at steady state.</li>
	<li>Repeat step 2.4 at least four times using elevated heater powers. Ensure that the wall temperatures corresponding to the heater powers used by steps 2.3 and 2.4 cover the Tw range determined by step 1.2.</li>
	<li>Calculate the Tw values scanned by the infrared thermography system using a number of selective thermal emissivity coefficients for the program that converts the infrared signals into temperature data.</li>
	<li>Compare the Tw data measured by the calibrated thermocouple and the infrared thermography system at the location corresponding to the thermocouple spot with the standard deviations evaluated.</li>
	<li>Select the thermal emissivity coefficient with the minimum standard deviation determined by step 2.7.</li>
	<li>Determine the maximum precision error for the infrared thermography system using the thermal emissivity coefficient determined by step 2.8.</li>
</ol>
3. Dynamic Balance of Rotating Rig
<ol>
	<li>Install the heat transfer test module, the infrared camera, the enveloping frame and all accessories on the rotating rig.</li>
	<li>Adjust the counterbalancing weight gradually until the running condition of the rotating rig satisfies the vibrational limitation for the infrared thermographic measurements to exhibit the stable thermal image on the computer display.</li>
</ol>
4. Evaluation of Heat Loss Coefficients
<ol>
	<li>Fill the coolant channel of the heat transfer test module with thermal insulation material.</li>
	<li>Install the filled test module on the rotating test rig by fitting the test module on the rotating platform and connecting the heater power supply and all the instrumental cables.</li>
	<li>Activate the data acquisition system to scan the temporal Tw variation at a heating power until the steady state condition is satisfied. Ensure that the temporal Tw variations during several successive scans are less than +0.3 K at each steady state condition.</li>
	<li>Record the heater power, steady-state Tw data and the corresponding ambient temperature, T&#8734;.</li>
	<li>Repeat steps 4.3 and 4.4 at least five times using different heating powers at a fixed rotating speed.</li>
	<li>Repeat steps 4.2 - 4.4 with at least five rotating speeds. Ensure that the test range of the rotating speed covers all the N values determined by step 1.2.</li>
	<li>Repeat steps 4.3 - 4.6 with a reversed rotating direction.</li>
	<li>Construct the plots of heat loss flux against wall-to-ambient temperature difference at each rotating speed.</li>
	<li>Correlate the heat loss coefficients as the functions of wall-to-ambient temperature difference, rotating speed and direction of rotation.</li>
	<li>Incorporate the heat loss correlation into the post data process program for Nu accountancy.</li>
</ol>
5. Baseline Heat Transfer Tests
<ol>
	<li>Perform heat transfer tests at the targeting Reynolds numbers at zero rotating speed (Ro = N = 0) by feeding coolant flows and heater powers to the test module. Ensure the supplied coolant mass flow rate is constantly adjusted in order to control Reynolds number at the flow entry plane at the targeting value.</li>
	<li>Record all the relevant raw data, including the steady state wall temperatures, fluid temperatures, heater powers, flow pressures and ambient pressures and temperatures, for subsequent data processing.</li>
	<li>Evaluate the local and area-averaged Nusselt numbers (Nu0) over the scanned static channel walls.</li>
</ol>
6. Rotating Heat Transfer Tests
<ol>
	<li>Install the on-line monitoring program to monitor the test conditions at the targeting Re and Ro.</li>
	<li>Feed the measured coolant mass flow rate, airflow pressure, rotating speed and fluid temperature at channel entrance into the monitoring program to calculate the instant Re and Ro.</li>
	<li>Record all the relevant raw data, such as rotating speed, heater power, airflow and ambient pressures, as well as the wall and fluid temperatures for subsequent data processing after the pre-defined steady-state condition is satisfied.</li>
	<li>Repeat steps 6.2 and 6.3 with at least four ascending or descending heater powers at a set of fixed Re and Ro. Ensure that the test Re and Ro fall within &#177;1% differences from the targeting values by adjusting the rotating speed or the coolant mass flow rate or both.</li>
	<li>Ensure that the heat transfer tests at each set of fixed Re and Ro with different heater powers are continuously performed as the development of buoyancy induced flows is associated with the &#34;history&#34; of the flow development.</li>
	<li>Repeat steps 6.4 and 6.5 with four or five targeting Reynolds numbers (Re) at a fixed rotation number (Ro). Ensure the rotating speed is appropriately adjusted at each test Re to control both Re and Ro at the targeting values within &#177;1% differences.</li>
	<li>Repeat step 6.6 using four or five targeting rotation numbers (Ro).</li>
	<li>Repeat steps 6.2 to 6.7 with reversed rotating direction.</li>
	<li>Evaluate the local and area-averaged Nusselt numbers (Nu) over the scanned rotating channel walls using a post data processing program.</li>
</ol>
7. Parametric Analysis
<ol>
	<li>Correlate the area-averaged Nusselt numbers (Nu0) collected from the static channel into the functions of Reynolds number.</li>
	<li>Evaluate the full-field local Nu/Nu0 ratios at each fixed Re and Ro tested with the area-averaged Nu/Nu0 ratios calculated.</li>
	<li>Verify the applicability of isolation Re effect by plotting the local and area-averaged Nu/Nu0 ratios obtained with different Re but at identical Ro.</li>
	<li>Disclose the isolated impacts of rotating buoyancy on heat transfer properties of the rotating test channel by plotting the area-averaged Nu/Nu0 ratios collected at the same Ro with different Re against Bu or density ratio (&#916;&#961;/&#961;). Ensure the preferable selection of Bu or &#916;&#961;/&#961; to construct this type of plot for obtaining the consistent data trend with a simple functional structure for heat transfer correlation.</li>
	<li>Extrapolate each Nu/Nu0 data trend collected at a fixed Ro but different Re into the limiting condition of Bu&#8594;0 or &#916;&#961;/&#961;&#8594;0.</li>
	<li>Collect all the extrapolated Nu/Nu0 results with Bu&#8594;0 or &#916;&#961;/&#961;&#8594;0 at all the tested Ro.</li>
	<li>Plot the extrapolated Nu/Nu0 results with vanished buoyancy interaction against Ro to disclose the uncoupled Coriolis force effects on the heat transfer properties.</li>
	<li>Correlate the test results collected by steps 7.4 and 7.7 into the functions of Ro and Bu.</li>
</ol>

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The authors have nothing to disclose.

While the endwall temperatures of a rotating channel are detected by an infrared thermography system, the fluid temperatures are measured by thermocouples. As the alternative magnetic field of an AC motor that drives a rotating rig induces electrical potential to interfere the thermocouple measurements, the DC motor must be adopted to drive a rotating test rig.
The fluid temperature distribution over the exit plane of a heated channel is not uniform. At least five thermocouples on the existing...

While thermodynamic laws dictate the improved specific power and thermal efficiency of a gas turbine engine by elevating the turbine entry temperature, several hot engine components, such as turbine blades, are prone to thermal damage. Internal cooling of a gas turbine rotor blade permits a turbine entry temperature in excess of the temperature limits of the creep resistance of the blade material. However, the configurations of the internal cooling channels must comply with the blade profile. In particular, the coolant rotates within the rotor blade. With such harsh thermal conditions for a running gas turbine rotor blade, an effective blade cooling scheme is crucial to ensure the structure&#39;s integrity. Thus, the local heat transfer properties for a rotating channel are important for the efficient usage of the limited coolant flow available. The acquisition of useful heat transfer data that are applicable to the design of the internal coolant passages at realistic engine conditions is of primary importance when an experimental method is developed for measuring the heat transfer properties of a simulated cooling passage inside a gas turbine rotor blade.
Rotation at a speed above 10,000 rpm considerably alters the cooling performance of a rotating channel inside a gas turbine rotor blade. The identification of engine conditions for such a rotating channel is permissible using the similarity law. With rotation, the dimensionless groups that control the transport phenomena inside a radially rotating channel can be revealed by deriving the flow equations relative to a rotating frame of reference. Morris1 has derived the momentum conservation equation of flow relative to a rotating frame of reference as:
<img alt="Equation 1" src="/files/ftp_upload/57630/57630eq1.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(1)
In equation (1), the local fluid velocity, v&#772;, with the position vector, r&#772;, relative to a frame of reference rotating at the angular velocity, &#969;, is affected by the Coriolis acceleration in terms of 2(&#969;&#215;v&#772;), the decoupled centripetal buoyancy force, &#946;(T-Tref)(&#969;&#215;&#969;&#215;r&#772;), the driven piezo-metric pressure gradient, <img alt="Equation 16" src="/files/ftp_upload/57630/57630eq16.jpg" />, and the fluid dynamic viscosity, &#957;. The referenced fluid density, &#961;ref, is referred to a pre-defined fluid reference temperature Tref, which is typical of the local fluid bulk temperature for experiments. If the irreversible conversion of mechanical energy into thermal energy is negligible, the energy conservation equation is reduced to:
<img alt="Equation 2" src="/files/ftp_upload/57630/57630eq2.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(2)
The first term of equation (2) is obtained by treating the specific enthalpy to be directly related to the local fluid temperature, T, via the constant specific heat, Cp. As the perturbation of fluid density caused by the variation of fluid temperature in a heated rotating channel provides considerable influence on the motion of fluids when it links with the centripetal acceleration in equation (1), the fluid velocity and temperature fields in an axially rotating channel are coupled. Also, both Coriolis and centripetal accelerations vary simultaneously as the rotating speed is adjusted. Thus, the effects of Coriolis force and rotating buoyancy on the fields of fluid velocity and temperature are naturally coupled.
Equations (1) and (2) in the dimensionless forms disclose the flow parameters that govern the heat convection in a rotating channel. With a basically uniform heat flux imposed on a rotating channel, the local fluid bulk temperature, Tb, increases linearly in the streamwise direction, s, from the reference inlet level, Tref. The local fluid bulk temperature is determined as Tref + &#964;s, where &#964; is the gradient of the fluid bulk temperature in the direction of flow. Substitutions of the following dimensionless parameters of:
<img alt="Equation 3" src="/files/ftp_upload/57630/57630eq3.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(3)
<img alt="Equation 4" src="/files/ftp_upload/57630/57630eq4.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(4)
<img alt="Equation 5" src="/files/ftp_upload/57630/57630eq5.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(5)
<img alt="Equation 6" src="/files/ftp_upload/57630/57630eq6.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(6)
<img alt="Equation 7" src="/files/ftp_upload/57630/57630eq7.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(7)
into equations (1) and (2), where Vmean, N and d respectively stand for the mean flow through velocity, rotating velocity and channel hydraulic diameter, the dimensionless flow momentum and energy equations are derived as equations (8) and (9) respectively.
<img alt="Equation 8" src="/files/ftp_upload/57630/57630eq8.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(8)
<img alt="Equation 9" src="/files/ftp_upload/57630/57630eq9.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(9)
Evidently, &#951; in equation (9) is a function of Re, Ro, and Bu = Ro2&#946;&#964;dR, which are respectively referred to as Reynolds, rotation and buoyancy numbers. The Rossby number that quantifies the ratio between inertial and Coriolis forces is equivalent to the inverse rotation number in equation (8).
When Tb is calculated as Tref + &#964;s in a rotating channel subject to a uniform heat flux, the &#964; value can be alternatively evaluated as Qf/(mCpL) in which Qf, m and L are the convective heating power, coolant mass flow rate and channel length, respectively. Thus, the dimensionless local fluid bulk temperature, &#951;b, is equal to s/d and the dimensionless temperature at channel wall, &#951;w, yields [(Tw-Tb)/Qf][mCp][L/d]+s/d. With the convective heat transfer rate defined as Qf/(Tw-Tb), the dimensionless wall-to-fluid temperature difference, &#951;w-&#951;b, is convertible into the local Nusselt number via equation (10) in which &#950; is the dimensionless shape function of heating area and channel sectional area.
<img alt="Equation 10" src="/files/ftp_upload/57630/57630eq10.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(10)
With a set of predefined geometries and the hydrodynamic and thermal boundary conditions, the dimensionless groups controlling the local Nusselt number of a rotating channel are identified as:
<img alt="Equation 11" src="/files/ftp_upload/57630/57630eq11.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(11)
<img alt="Equation 12" src="/files/ftp_upload/57630/57630eq12.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(12)
<img alt="Equation 13" src="/files/ftp_upload/57630/57630eq13.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(13)
With experimental tests, the adjustment of rotating speed, N, for varying Ro to generate the heat transfer data at different strengths of Coriolis forces inevitably changes the centripetal acceleration, and thus, the relative strength of rotating buoyancy. Moreover, a set of heat transfer data collected from a rotating channel is always subject to a finite degree of rotating buoyancy effect. To disclose the individual effects of Coriolis-force and buoyancy on the heat transfer performance of a rotating channel requires the uncoupling of the Ro and Bu effects on Nu properties through the post data processing procedure that is inclusive in the present experimental method.
The engine and laboratory flow conditions for a rotating channel inside a gas turbine rotor blade can be specified by the ranges of Re, Ro and Bu. The typical engine conditions for the coolant flow through a gas turbine rotor blade, as well as the construction and commissioning of the rotating test facility that allowed experiments to be performed near the actual engine conditions was reported by Morris2. Based on the realistic engine conditions summarized by Morris2, Figure 1 constructs the realistic operating conditions in terms of Re, Ro and Bu ranges for a rotating coolant channel in a gas turbine rotor blade. In Figure 1, the indication of an engine&#39;s worst condition is referred to as the engine running condition at the highest rotor speed and the highest density ratio. In Figure 1, the lower limit and worst engine operating conditions respectively emerge at the lowest and highest engine speeds. It is extremely difficult to measure the full-field Nu distribution of a rotating channel running at a real engine speed between 5000 and 20,000 rpm. However, based on the similarity law, laboratory-scale tests have been conducted at reduced rotating speeds but with several attempts to provide a full coverage of the real-engine Re, Ro and Bu ranges. As an innovative experimental method, the NASA HOST program3,4,5,6 adopted the high-pressure tests for increasing the fluid densities at the predefined Re in order to extend the Ro range by reducing the mean fluid velocity. In this regard, the specific relationships between Re, Ro and Bu for an ideal gas with a gas constant, Rc, and viscosity, &#956;, are related as:
<img alt="Equation 14" src="/files/ftp_upload/57630/57630eq14.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(14)
<img alt="Equation 15" src="/files/ftp_upload/57630/57630eq15.jpg" />&#160;&#160;&#160;&#160;&#160;&#160;(15)
To bring the laboratory conditions into the nominal correspondence with engine conditions seen in Figure 1, the rotating speed, N, coolant pressure, P, channel hydraulic diameter, d, rotating radius, R, and wall-to-fluid temperature difference, Tw-Tb, need to be controlled for matching the realistic Re, Ro and Bu ranges. Clearly, one of the most effective approaches to extend the Ro range is to increase channel hydraulic diameter, as Ro is proportional to d2. As the laboratory heat transfer test at realistic N is extremely difficult, the coolant pressure, P, is technically easier to be raised for extending Ro range; even if Ro is only proportional to P. Based on this theoretical background, the design philosophy of the present experimental method is to increase Ro by pressurizing the rotating test channel using the maximum channel hydraulic diameter allowed to fit into the rotating rig. Having increased the Ro range, the range of Bu is accordingly extended as Bu is proportional to Ro2. In Figure 1, the laboratory test conditions adopted to generate the heat transfer data of rotating channels are also included3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29. As indicated in Figure 1, the coverage of realistic engine conditions by the available heat transfer data is still limited, especially for the required Bu range. The open and the colored solid symbols depicted in Figure 1 are the pointed and full-field heat transfer experiments, respectively. As collected in Figure 1, most of the heat transfer data with cooling applications to gas turbine rotor blades1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26 are point measurements using the thermocouple method. The wall conduction effects on measuring the wall conductive heat flux and the temperatures at fluid-wall interfaces undermine the quality of heat transfer data converted from the thermocouple measurements. Also, the heat transfer measurements1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26 using the thermocouple method cannot detect the two-dimensional heat transfer variations over a rotating surface. With the present experimental method29,30,31,32, the detection of full-field Nusselt number distributions over the rotating channel wall is permissible. The minimization of wall conduction effect using 0.1 mm thick stainless-steel foils with Biot numbers &#62;&#62;1 to generate the heating power by the present experimental method permits the one-dimensional heat conduction from the heating foil to the coolant flow. In particular, the acquisition of full-field heat transfer data involving both Ro and Bu effects is not permissible using the transient liquid crystal technique and the thermocouple method. With the current steady-state liquid crystal thermography method19, the detectable temperature range of 35-55 &#176;C disables the generation of heat transfer data with realistic density ratios.
Using the flow parameters governing the heat convection in a rotating channel to demonstrate that the full coverage of realistic engine conditions seen in Figure 1 has not yet been achieved, so the need for acquiring the full-field heat transfer data at realistic engine conditions has been continuously urged. The present experimental method enables the generation of full-field heat transfer with both Coriolis-force and rotating-buoyancy effects detected. The protocols are aimed at assisting the investigators to devise an experimental strategy relevant to the realistic full-field heat transfer measurement of a rotating channel. Along with the method of parametric analysis that is unique to the present experimental method, the generation of heat transfer correlation for assessing the isolated and interdependent Ro and Bu effects on Nu is permitted.
The article illustrates an experimental method aimed at generating the two-dimensional heat transfer data of a rotating channel with flow conditions similar to the realistic gas turbine engine conditions but operating at much lower rotating speeds in the laboratories. The method developed to select the rotating speed, the hydraulic diameter of test channel and the range of wall-to-fluid temperature differences for acquiring the heat transfer data at realistic engine conditions is illustrated in the introduction. The calibration tests for the infrared thermography system, the heat loss calibration tests and the operation of the rotating heat transfer test rig are shown. The factors causing the significant uncertainties for heat transfer measurements and the procedures for decoupling the Coriolis-force and buoyancy effects on the heat transfer properties of a rotating channel are described in the article with the selective results to demonstrate the present experimental method.

<table><tbody><tr><td>Rotating test rig</td><td>In-house made</td><td></td><td>Design by this research group</td></tr><tr><td>Heat transfer test module</td><td>In-house made</td><td></td><td>Design by this research group</td></tr><tr><td>Mass flow meter</td><td>Eldride Product, Inc.</td><td>3100301-01-01&lt;br/&gt; 359-1007</td><td></td></tr><tr><td>Infrared thermography system</td><td>NEC P384A-8</td><td>3100401-04&lt;br/&gt; 3127A-4</td><td></td></tr><tr><td>Instrumentation slip ring</td><td>Michigan Scientific SR36M</td><td>3100506-62&lt;br/&gt; 3553-372</td><td></td></tr></tbody></table>

uncoupling coriolis force and rotating buoyancy effects on full-field heat transfer properties of a rotating channel

An experimental method for exploring the heat transfer characteristics of an axially rotating channel is proposed. The governing flow parameters that characterize the transport phenomena in a rotating channel are identified via the parametric analysis of the momentum and energy equations referring to a rotating frame of reference. Based on these dimensionless flow equations, an experimental strategy that links the design of the test module, the experimental program and the data analysis is formulated with the attempt to reveal the isolated Coriolis-force and buoyancy effects on heat transfer performances. The effects of Coriolis force and rotating buoyancy are illustrated using the selective results measured from rotating channels with various geometries. While the Coriolis-force and rotating-buoyancy impacts share several common features among the various rotating channels, the unique heat transfer signatures are found in association with the flow direction, the channel shape and the arrangement of heat transfer enhancement devices. Regardless of the flow configurations of the rotating channels, the presented experimental method enables the development of physically consistent heat transfer correlations that permit the evaluation of isolated and interdependent Coriolis-force and rotating-buoyancy effects on the heat transfer properties of rotating channels.

Realistic operating conditions for the internal coolant flows inside a rotating gas turbine blade in terms of Re, Ro and Bu are compared with the emulated laboratory conditions in Figure 1. The data points fall in the realistic engine conditions using the present experimental method summarized in the protocols11,14,17,20,<sup c...

Watch this Scientific Journal Video about Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel at JoVE.com

Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

Here, we present an experimental method for decoupling the interdependent Coriolis-force and rotating-buoyancy effects on full-field heat transfer distributions of a rotating channel.

An experimental method for exploring the heat transfer characteristics of an axially rotating channel is proposed. The governing flow parameters that ...

uncoupling-coriolis-force-rotating-buoyancy-effects-on-full-field

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Unit 1

Equilibrium of a Rigid Body

Open Channel Flow

Dynamics of Rotational Motions

SCIENTISTS IN ACTION

8.1K Views.  National Cheng Kung University. Here, we present an experimental method for decoupling the interdependent Coriolis-force and rotating-buoyancy effects on full-field heat transfer distributions of a rotating channel.

Video: Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

Magnetically Induced Rotating Rayleigh-Taylor Instability

Classical techniques for investigating the Rayleigh-Taylor instability include using compressed gasses1, rocketry2 or linear electric motors3 to reverse the effective direction of gravity, and accelerate the lighter fluid toward the denser fluid. Other authorse.g. 4,5,6 have separated a gravitationally unstable stratification with a barrier that is removed to initiate the flow. However, the parabolic initial interface in the case of a rotating stratification imposes significant technical difficulties experimentally. We wish to be able to spin-up the stratification into solid-body rotation and only then initiate the flow in order to investigate the effects of rotation upon the Rayleigh-Taylor instability. The approach we have adopted here is to use the magnetic field of a superconducting magnet to manipulate the effective weight of the two liquids to initiate the flow. We create a gravitationally stable two-layer stratification using standard flotation techniques. The upper layer is less dense than the lower layer and so the system is Rayleigh-Taylor stable. This stratification is then spun-up until both layers are in solid-body rotation and a parabolic interface is observed. These experiments use fluids with low magnetic susceptibility, |&#967;| ~ 10-6 - 10-5, compared to a ferrofluids. The dominant effect of the magnetic field applies a body-force to each layer changing the effective weight. The upper layer is weakly paramagnetic while the lower layer is weakly diamagnetic. When the magnetic field is applied, the lower layer is repelled from the magnet while the upper layer is attracted towards the magnet. A Rayleigh-Taylor instability is achieved with application of a high gradient magnetic field. We further observed that increasing the dynamic viscosity of the fluid in each layer, increases the length-scale of the instability.

We present a protocol for preparing a two-layer density-stratified liquid that can be spun-up into solid body rotation and subsequently induced into Rayleigh-Taylor instability by applying a gradient magnetic field.

Classical techniques for investigating the Rayleigh-Taylor instability include using compressed gasses1, rocketry2 or linear electric motors3 to reverse ...

Pool-Boiling Heat-Transfer Enhancement on Cylindrical Surfaces with Hybrid Wettable Patterns

In this study, pool-boiling heat-transfer experiments were performed to investigate the effect of the number of interlines and the orientation of the hybrid wettable pattern. Hybrid wettable patterns were produced by coating superhydrophilic SiO2 on a masked, hydrophobic, cylindrical copper surface. Using de-ionized (DI) water as the working fluid, pool-boiling heat-transfer studies were conducted on the different surface-treated copper cylinders of a 25-mm diameter and a 40-mm length. The experimental results showed that the number of interlines and the orientation of the hybrid wettable pattern influenced the wall superheat and the HTC. By increasing the number of interlines, the HTC was enhanced when compared to the plain surface. Images obtained from the charge-coupled device (CCD) camera indicated that more bubbles formed on the interlines as compared to other parts. The hybrid wettable pattern with the lowermost section being hydrophobic gave the best heat-transfer coefficient (HTC). The experimental results indicated that the bubble dynamics of the surface is an important factor that determines the nucleate boiling.

Pool-boiling heat-transfer experiments were carried out to observe the effects of hybrid wettable patterns on the heat-transfer coefficient (HTC). The parameters of investigation are the number of interlines and the pattern orientation of the modified wettable surface.

In this study, pool-boiling heat-transfer experiments were performed to investigate the effect of the number of interlines and the orientation of the ...

Irrotational Flow

Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:
<img alt="Equation 1" src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;" />
must be satisfied, along with similar conditions for other axes:
<img alt="Equation 2" src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;" />
A uniform flow, where u is constant and v and w are both zero, satisfies these conditions and serves as a simple example of irrotational flow. However, when a solid object is introduced, irrotationality only persists far from the object. Near the boundary, the velocity increases sharply from zero (due to the no-slip condition) to the freestream value, creating significant shear stress and a boundary layer in viscous fluids.
When fluid enters a pipe from a large reservoir, the initial flow is nearly uniform, resulting in an irrotational core where the velocity distribution is flat. As the fluid moves downstream, a boundary layer forms along the pipe walls due to viscosity, causing the velocity profile to develop gradients that ultimately lead to a fully developed parabolic profile. In this fully developed region, rotational effects dominate near the walls, while a central core may remain approximately irrotational.
In the initial irrotational region of pipe flow, Bernoulli&#39;s equation
<img alt="Equation 3" src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;" />
can be applied between points along a streamline as the conditions of incompressibility, inviscid flow, and irrotationality hold. However, as the boundary layer grows and rotational effects become significant near the walls, the assumptions of Bernoulli&#39;s equation break down. This means that Bernoulli&#39;s equation is only applicable in the entrance region and the irrotational core, but not in the fully developed, rotational region where viscosity effects dominate.

Przepływ irrotacyjny

Irrotational flow refers to the movement of fluid particles that do not rotate along the flow path, characterized by a velocity field with zero curl and ...

Differential Analysis of Fluid Flow

KEY TERMS AND CONCEPTS

Mechanism of heat transfer

Understanding heat transfer mechanisms is essential for understanding how our bodies maintain balance in different environmental conditions. When the environment is thermoneutral, the body is in a state of balance, neither using nor releasing energy to maintain its core temperature. However, when the environment is not thermoneutral, the body employs four heat transfer mechanisms to maintain homeostasis: conduction, convection, evaporation, and radiation. These mechanisms facilitate heat transfer from higher concentration to lower concentration regions and are fascinating and essential to our understanding of human physiology.
Conduction, a heat transfer mechanism, is a process we encounter daily. It&#39;s the transfer of heat between two objects in direct contact. In humans, this occurs when the skin contacts an object with a different temperature. For example, when a nurse places an ice pack on a patient with a fever, heat transfers from the warmer body to the colder ice pack, thereby reducing the patient&#39;s body temperature. This practical application of conduction demonstrates its relevance and importance in our understanding of thermoregulation.
Convection, another heat transfer mechanism, is a process we can observe in our surroundings. It involves the transfer of heat away from the body through air movement. It occurs when heat from the skin warms the surrounding air, which then rises and is replaced by cooler air that warms up. An example is using a fan to cool down; the fan speeds up air movement around the body, enhancing heat loss through convection. This real-life example of convection helps us visualize and understand the concept better.
Evaporation is when a liquid converts into a gas, releasing heat from the body. It is a primary method by which the body loses heat. For instance, during physical activity, humans perspire, and as the sweat evaporates, it cools the body.
Radiation is the emission of heat from the surface of one object to another without direct contact between them. For example, the body can radiate heat to the cooler surrounding environment or absorb heat from the sun, increasing body temperature without physical contact between the body and the heat source. For example, radiation is when our body feels warm standing near a bonfire, even without touching it.

Mechanizm wymiany ciepła

Understanding heat transfer mechanisms is essential for understanding how our bodies maintain balance in different environmental conditions. When the ...

Nursing

Vital Signs: Temperature

Mechanizm przenoszenia ciepła

Hydrostatic Pressure Force on a Curved Surface

Hydrostatic pressure on curved surfaces is a fundamental concept in fluid mechanics with broad applications in the civil engineering field. When fluid is in contact with a curved surface, as in a reservoir, dam, or storage tank, it exerts pressure that varies in magnitude and direction along the curved surface. To assess the total hydrostatic force exerted by the fluid on a curved structure, engineers typically isolate the fluid volume adjacent to the surface and analyze the forces acting on imaginary horizontal and vertical planes intersecting this volume.
The horizontal and vertical forces result from the fluid pressure acting perpendicular to each plane, and their magnitudes depend on the fluid&#39;s specific weight and depth. Additionally, the self-weight of the fluid within the isolated volume acts through its center of gravity, contributing to the overall force on the curved surface. By summing the squares of the horizontal and vertical components and finding the square root, engineers can calculate the total hydrostatic force on the curved surface.
Understanding these forces is essential for designing safe and efficient hydraulic structures, such as dams, pipelines, and containment vessels. Properly assessing hydrostatic pressure on curved surfaces helps engineers ensure structural stability under varying fluid loads, contributing to the long-term durability and safety of essential infrastructure.

Siła ciśnienia hydrostatycznego działająca na zakrzywioną powierzchnię

Consider a swimming pool with a fluid of a known specific weight having a curved surface at the bottom.
If the volume around the curved surface is ...

Fluid Statics

Steady, Laminar Flow in Circular Tubes

Hagen-Poiseuille flow describes a viscous fluid&#39;s steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,&#952;,z) are used to describe the radial (r), angular (&#952;), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial, meaning the velocity vector points only along the z-axis and varies solely with the radial distance r from the center of the tube.
The governing equations for fluid motion, the Navier-Stokes equations, simplify under these conditions. Without angular or radial velocity components, the axial velocity vz(r) emerges as a function of r alone. Solving the reduced Navier-Stokes equations yields a parabolic velocity profile:
<img alt="Equation 1" src="/files/ftp_upload/18065/18065_Equation 1.svg" style="text-align: center; height: 27px;" />
This profile demonstrates that viscous drag decreases the fluid velocity from the maximum at the center to zero at the tube wall. This zero velocity at the boundary, known as the no-slip condition, results from the friction between the fluid and the tube wall. Consequently, the flow is laminar, with fluid particles moving in parallel layers without lateral mixing.
Integrating the velocity profile over the cross-sectional area of the tube yields the volumetric flow rate Q, known as Poiseuille&#39;s law:
<img alt="Equation 2" src="/files/ftp_upload/18065/18065_Equation 2.svg" style="text-align: center; height: 27px;" />
Where &#916;P is the pressure difference across the length L &#160;of the tube, and &#956; is the fluid&#39;s dynamic viscosity. This relation shows that Q is highly sensitive to the tube&#39;s radius, increasing with the fourth power of R. It follows that even a slight increase in radius significantly boosts the flow rate, a principle critical in fluid transport applications.
From Poiseuille&#39;s law, the mean velocity V of the fluid can be expressed as:
<img alt="Equation 3" src="/files/ftp_upload/18065/18065_Equation 3.svg" style="text-align: center; height: 27px;" />
This average velocity is precisely half of the maximum velocity at the tube&#39;s center, confirming the parabolic velocity distribution and demonstrating the predictable, layered nature of laminar flow in a cylindrical tube.

Stały, laminarny przepływ w okrągłych rurach

Consider steady, incompressible flow through a straight cylindrical tube of constant radius R, known as Hagen&#8211;Poiseuille flow.
The cylindrical ...

Mechanisms of Heat Transfer I

Just as interesting as the effects of heat transfer on a system are the methods by which the heat transfer occur. Whenever there is a temperature difference, heat transfer occurs. It may occur rapidly, such as through a cooking pan, or slowly, such as through the walls of a picnic ice box. So many processes involve heat transfer that it is hard to imagine a situation where no heat transfer occurs. Yet, every heat transfer takes place by only three methods: conduction, convection, and radiation.
For example, heat transfer occurs in a fireplace by all three methods: conduction, convection, and radiation. Radiation is responsible for most of the heat transferred into the room. Heat transfer also occurs through conduction into the room, but much more slowly. Heat transfer by convection also occurs through cold air entering the room around the windows and hot air leaving the room by rising up the chimney.
Conduction is heat transfer through stationary matter by physical contact. The matter is stationary only on a macroscopic scale as thermal motion of the atoms and molecules occurs at any temperature above absolute zero. Heat transferred from a stove&#8217;s burner through the bottom of a pan to the food in the pan is transferred by conduction.

Mechanizmy wymiany ciepła I

Just as interesting as the effects of heat transfer on a system are the methods by which the heat transfer occur. Whenever there is a temperature ...

Temperature and Heat

Mechanisms of Heat Transfer II

In convection, thermal energy is carried by the large-scale flow of matter. Ocean currents and large-scale atmospheric circulation, which result from the buoyancy of warm air and water, transfer hot air from the tropics toward the poles and cold air from the poles toward the tropics. The Earth&#8217;s rotation interacts with those flows, causing the observed eastward flow of air in the temperate zones. Convection dominates heat transfer by air, and the amount of available space for the airflow determines whether the air transfers heat rapidly or slowly. There is little heat transfer in a space filled with air with a small amount of other material that prevents flow.
The space between the Earth and the Sun is largely empty, so the Sun warms us without any possibility of heat transfer by convection or conduction. In this example, heat is transferred by radiation. That is, the hot body emits electromagnetic waves that are absorbed by the skin. No medium is required for the electromagnetic waves to propagate.
Most of the heat transfer from a fire to the observers occurs through infrared radiation. The visible light transfers relatively little thermal energy. The energy of electromagnetic radiation varies over a wide range depending on the wavelength, with shorter wavelengths (or higher frequencies) corresponding to higher energy. As more heat is radiated at higher temperatures, higher temperatures produce more intensity at every wavelength, especially at shorter wavelengths. In visible light, the wavelength determines the color&#8212;red has the longest wavelength and violet the shortest. Therefore, temperature changes are accompanied by color changes. For example, the electric heating element on a stove glows from red to orange, while the higher-temperature steel in a blast furnace glows from yellow to white.

Mechanizmy wymiany ciepła II

In convection, thermal energy is carried by the large-scale flow of matter. Ocean currents and large-scale atmospheric circulation, which result from the ...

Coriolis Force

An accelerating particle experiences a force equal to the mass multiplied by the acceleration in an inertial frame of reference. Consider a particle in a non-inertial frame of reference, such as a sliding ball on a rotating table. The acceleration of the ball in this rotating reference frame is different than in the intertial frame, which modifies its equation of motion. The fictitious forces acting additionally on a rotating frame of reference alter Newton&#39;s Second Law expression. Centripetal and Coriolis forces are the fictitious or pseudo forces that act on a particle in a non-inertial frame of reference.
The discovery of the Coriolis force dates back to 1651 when Italian military officers reported that, during artillery practice, the cannon balls always landed to the right of where the calculations predicted. The Coriolis force is named after a French mathematician Gaspard Gustave Coriolis (1792&#8211;1843) for his work in the field of rotating reference frames.
The Coriolis force is proportional to the cross-product of the rotating frame&#39;s angular velocity vector and the object&#39;s velocity vector. Its direction can be estimated using the right-hand rule for the cross-product. Suppose the index finger and thumb are directed toward the object&#39;s velocity and the rotating frame&#39;s angular velocity, respectively. The middle finger points opposite to the Coriolis force, as indicated by the negative sign in the Coriolis force expression. The work done by Coriolis force is zero since it always acts perpendicular to the object&#39;s motion.
Due to the Earth&#39;s rotation, the Coriolis force tends to deflect moving bodies toward the right in the northern hemisphere and toward the left in the southern hemisphere. As a result, long-range snipers aim to the left of their target in the northern hemisphere.
The Coriolis force also results in cyclone formation. In the northern hemisphere, air rushing out of a high-pressure system swirls clockwise around the pressure center. Similarly, air flowing into a low-pressure region swirls counterclockwise. The turn direction of the swirling is opposite in the southern hemisphere due to the Coriolis force. The Coriolis force is zero at the equator, and as a result, no cyclones occur at the equatorial axis.

Siła Coriolisa

An accelerating particle experiences a force equal to the mass multiplied by the acceleration in an inertial frame of reference. Consider a particle in a ...

Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

Podsumowanie

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Rozdziały w tym wideo

Magnetically Induced Rotating Rayleigh-Taylor Instability

Pool-Boiling Heat-Transfer Enhancement on Cylindrical Surfaces with Hybrid Wettable Patterns

Przepływ irrotacyjny

Mechanizm wymiany ciepła

Mechanizm wymiany ciepła

Siła ciśnienia hydrostatycznego działająca na zakrzywioną powierzchnię

Stały, laminarny przepływ w okrągłych rurach

Mechanizmy wymiany ciepła I

Mechanizmy wymiany ciepła II

Siła Coriolisa