Constant Reynolds Number Research Articles

Investigations into transient wakes behind a custom airfoil undergoing pitching motion

The wake dynamics behind a sinusoidally pitching custom designed symmetrical airfoil is analyzed computationally. The effect of pitching frequency and pitching amplitude is investigated independently at a constant Reynolds number. The wake patterns are found to exhibit a transition from von Karman to reverse von Karman vortex street and from symmetric reverse von Karman vortex street to asymmetric deflected wake as established. For laminar flow under investigation, the contours of time-varying velocity fluctuations are presented. The mean profiles of velocity and the fluctuations at different locations in the wake show that the thrust wakes are generated at the expense of energy of the surrounding fluid. The temporal behavior of the flow field is analyzed using the phase plot. The time series plots of force coefficients show a periodic loading on the airfoil. Proper orthogonal decomposition (POD) analysis of drag, neutral and thrust wakes displayed the evolution of coherent structures in the flow field. The evolution of these structures portray the transitions in the wake of the airfoil. The merging of coherent structures in an asymmetric deflected wake turns out to be an efficient way of identifying the location of deflection.

Techno-economic evaluation of different operating scenarios for indigenous and imported coal blends and biomass co-firing on supercritical coal fired power plant performance

Pakistan has enormous coal and biomass resources, however, it imports coal from other parts of the world. Whereas, utilization of indigenous coal and biomass may be beneficial for the economy and local energy security. Hence, this study focuses on the likely effects of using coal blends and biomass co-firing with indigenous and imported coal on the performance of a supercritical power plant. A coal fired power plant is modeled using Aspen plus. Three different case studies are considered, which are; constant heat input (CHI), constant fuel input (CFI), and constant Reynold number (CRN). Conclusively, coal blending has resulted in reduced emissions than the use of low rank coal. Further, biomass co-firing has shown reduced emissions. For the CHI case, efficiency remains the same for all fuels. The highest efficiency in the CFI case is for South African coal and the lowest is for biomass. In the CRN case, South African coal has maximum efficiency while Thar coal has the minimum efficiency. Moreover, economic analysis of this study proposes that native resources must be used, provided the technical issues have resolved. Therefore, this study resulted in the comparison and applicability of different coal blends and renewable biomass fuel in a coal fired power plant.

An experimental study on heat transfer enhancement and flow characteristics of a tube with plain, perforated and dimpled twisted tape inserts

In order to increase the performance of a heat exchanger flowing ethylene glycol (EG) and water (W) mixtures, the effects of different twisted tapes on heat transfer and flow characteristics in a heat exchanger tube is experimentally investigated. Material of the twisted tapes and the tube is selected as stainless steel. Three different volumetric ratios of EG and W mixtures are used with the volumetric ratio of (0:100), (20:80) and (40:60). The twisted tapes having twist ratio of 5.88 are perforated and dimpled with pitch ratio of (Pp/y and Pd/y) of 0.25, 0.5 and 1.0. The experiments are conducted under constant heat flux and Reynold number ranging from 5217 to 22,754. The experiments for both the smooth tube and the typical twisted tape inserted tube are validated with the literature. The results indicated that the use of twisted tape inserts enhances the heat transfer performance. Experimental results revealed that novelty dimpled twisted tape insert provides higher heat transfer performance than the others. Adding the ethylene glycol on pure water has not major role on increase in the friction factor. As the amount of ethylene glycol in the mixture increases, thermo hydraulic performance decreases. The best thermo-hydraulic performance results are obtained as 1.42, 1.17 and 1.05 for dimpled twisted tape (Pd/y = 0.25) inserted tube at the lowest Reynolds numbers for the fluids which are (0:100), (20:80) and (40:60), respectively.

Incompressible SPH simulation of flow past horizontal cylinder between plane wall and free surface

Flow past a circular cylinder located between free-surface and wall boundaries is modelled numerically using non-uniform particle size incompressible smoothed particle hydrodynamics (SPH). Several enhancements including Rhie and Chow interpolation, colour function-based free surface tracking, and modified particle shifting are introduced to increase the accuracy and efficiency of the method. A parameter study at constant Reynolds number, Re=150, examines the influence of Froude number (Fr=0.2−0.6), submergence ratio (H∕D=0.5−1.0, where H is the distance between the apex of the cylinder and undisturbed free surface, and D is the cylinder diameter), and bottom gap ratio (G∕D=1.0−5.0, where G is the distance between the base of the cylinder and the bed) on the ambient free-surface flow pattern and hydrodynamic force on the cylinder. It is found that the vortex shedding pattern depends on all three parameters. As the Froude number increases for fixed submergence ratio and bottom gap ratio, vortex shedding is eventually suppressed. As the submergence and bottom gap ratios increase, the threshold of vortex shedding suppression shifts to higher values of Froude number. The mean drag coefficient depends on the submergence ratio and bottom gap ratio but is independent of Froude number. Meanwhile, the lift coefficient depends on submergence ratio and Froude number but is independent of bottom gap ratio. Spectral analysis of force and free-surface elevation time signals shows that the free-surface deformation and lift force are closely related during the vortex shedding regime.

An estimation method for the fuel burn and other performance characteristics of civil transport aircraft in the cruise. Part 1 fundamental quantities and governing relations for a general atmosphere

AbstractThis paper is one of a series addressing the need for simple, yet accurate, methods for the estimation of cruise fuel burn and other important aircraft performance parameters. Here, a previously published, constant Reynolds number model for turbofan-powered, civil transport aircraft is extended to include Reynolds number effects. Provided the variation of temperature with pressure is known, the method is applicable to flight in any atmospheric conditions. For a given aircraft, cruising in a given atmosphere, there is a single Mach number and Flight Level pair, at which the fuel burn per unit distance travelled through the air has an absolute minimum value. Both these quantities depend upon the Reynolds number, which, in turn, depends upon the aircraft weight and the atmospheric vertical temperature profile. Simple, explicit expressions are developed for all parameters at the optimum condition. These are shown to be in close agreement with numerical solutions of the governing equations. It is found that typical operational mass and temperature profile variations can change cruise fuel burn rate by several percent. In the International Standard Atmosphere, when the speed and altitude deviate from their optimum values, the fuel burn penalty is reduced slightly relative to the constant Reynolds number case. By way of example, the method is used to estimate the minimum fuel, speed-versus-height trajectory for cruise in a realistic atmosphere.For each aircraft, cruise fuel burn is found to be governed by six independent parameters. All are constants. Two are simple, involving only size and weight, whereas four are complex and must be determined by either theoretical, or empirical, means. The estimation of these quantities will be considered in Part 2.

Geometric optimization of a lid-driven cavity with two rectangular intrusions under mixed convection heat transfer: A numerical investigation motivated by constructal design

The present work aims at the geometric optimization, by means of constructal design and exhaustive search, of two rectangular fins inserted in a lid-driven cavity subjected to unsteady, incompressible, laminar, two-dimensional mixed convective flow with stable stratification. The main purpose is to maximize the dimensionless heat transfer rate (q˜′) and time and spatial-averaged Nusselt number (NuH¯). The domain presents three constraints: cross-sectional areas for each fin and total area of the cavity. Two degrees of freedom are investigated, height/length ratios of the left and right rectangular fin (H1/L1 and H2/L2), considering three different fraction areas for fins and two different Richardson numbers Ri = 0.1 and 1.0, representing two conditions for mixed convective flow. All cases have constant Reynolds and Prandtl numbers (ReH = 400 and Pr = 6.0). The conservation equations of mass, momentum, and energy are solved using the Finite Volume Method. Recommendations for fins shapes were strongly affected by the performance indicators chosen. Results also indicated that asymmetric configurations for the fins with different fraction areas for each fin led to the best thermal performance. Moreover, the optimal shapes and the effect of degrees of freedom over performance indicators were also affected by the Richardson numbers investigated.

Centerline heat transfer coefficient distributions of synthetic jets impingement cooling

The paper presents the results of experimental investigations of heat transfer coefficients, velocity and pressure distributions of synthetic jets for two cases: constant Reynolds number and variable Stokes number and vice versa. Measurements were made for synthetic jet Reynolds number in the range of Re=5440–11,475, dimensionless stroke length L0=7.49–38.5 and dimensionless jet-to-plate spacing x/d = 0–20. The synthetic jet time-averaged and momentum velocity as well as the pressure for each case are presented. The centerline turbulence intensity and the Euler number are also reported. The centerline heat transfer coefficient distributions are compared with the distributions of the various velocity and pressure parameters. A new correlation between the local Nusselt number and the local Reynolds number is presented. Under synthetic jet impingement cooling the maximum of the stagnation point heat transfer coefficient appears mostly at axial distance of x/d ≈ 5. The location of the maximum Nusselt number at x/d ≈ 5 is caused by the modification of the heat transfer distribution due to the recirculation of hot air. With the decrease of the heat power released into the control volume by the impingement plate (heater) this modifying effects becomes less pronounced thus the maximum of the heat transfer coefficient shifts towards the location of about x/d = 2. The maxima of various flow parameters are generally observed at about x/d = 2 what was also reported by other independent researchers. Thus, it may be concluded that the purely mechanical phenomena does not full coincide with thermal ones in the case of synthetic jet impingement cooling.

Reynolds-Averaged Simulation of the Fully Developed Turbulent Drag Reduction Flow in Concentric Annuli

AbstractReynolds-averaged modeling is performed for polymer-induced drag reduction (DR) fluid at the fully developed turbulent regime in a concentric annulus by using the commercial code, ansys-fluent. The numerical approach adopted in this study relies on a modified k–ε–v2¯–f model to characterize the turbulence and the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model to represent the rheological behavior of the polymer solution. The near-wall axial velocity, Reynolds stress, and turbulent kinetic energy (TKE) budget near both walls of the annulus (fixed radius ratio of 0.4) are compared in detail at a constant Reynolds number (Re=10,587) and various rheological parameters (Weissenberg number We in the range of 1–7 and the maximum polymer elongation L = 30 and 100). Current simulation has predicted the redistributions of turbulent statistics in the annulus, where the two turbulent boundary layers (TBLs) of the DR flow differ more compared to those of its Newtonian counterpart. The difference is also found to be highly dependent on the rheological properties of the viscoelastic fluid.

A Study on Aerodynamic Behavior of Subsonic UAVs' Wing Sections with Flaps

In this study, it is aimed to assess the aerodynamic and flight effects of the flap design on an airfoil. For this purpose, NACA 4415 type wing profile, which can also be used in unmanned aerial vehicles (UAVs), is selected. The original design and the +5-degree flapped design which has constant other design features are compared. Assessments are performed under constant Reynolds numbers and an angle of attack between 0-10 degrees with a 1-degree interval. Analyses are made using the open-source software XFLR5. For the flapped design is named NACA 4415-2, some basic aerodynamic performance parameters such as coefficient of drag (CD), coefficient of lift (CL) coefficient of pressure (Cp) maximum lift coefficient (Clmax) and minimum stall velocity (Vstall) have been observed. According to results, when the flap with 5o is added to the airfoil, it has been observed that the CL and Lift force of the original design of the airfoil increase significantly, CD of the airfoil increase partially, and the pressure coefficient tends to decrease significantly. Furthermore, it has been observed that while the minimum stall velocity has decreased, Clmax values increased.

Hydrothermal, energy and emissions characteristics assessment of impinging flame jet on an oval flat tube

The flow, heat transfer and emissions features of a methane–air premixed impinging flame jet on an oval flat tube involving both smooth and concave surfaces are numerically investigated. The effects of important parameters such as Reynolds number (Re = 400, 800, 1200, 1600 and 2000) and nozzle-to-tube distance (H/D = 3, 5 and 8) on streamlines, pressure coefficient temperature field, surface heat flux, energy efficiency and pollutants formation are evaluated. Results reveal that at higher Reynolds number, a larger recirculation zone beyond the concave surface is generated, the flame direction becomes horizontal and maximum temperature is shifted far away from stagnation point. The highest value of pressure coefficient is found at Re = 400 occurring on stagnation point. Under constant Reynolds number, the maximum temperature value belongs to configuration with less H/D in which the thermal boundary layer becomes narrower. Except for Re = 2000, the highest heat flux values are associated with stagnation point, while a considerable off-stagnation heat flux is detected for Re = 2000. More amount of energy releases for less H/D at lower Reynolds number, while an inverse behavior is observed at higher ones. Simulations show that the best energy efficiency is found at Re = 400 and H/D = 3. The production of CO species is of high remarkable value at Re = 400 when the space between nozzle and tube is reduced. The highest value of NOx production is distinguished at the highest H/D and lowest Re (i.e., H/D = 8 and Re = 400), while the concentration of UHC remarkably increases with rising Reynolds number, especially for higher H/D.

Parametric evaluation of a hydrofoil-shaped sidewall rib-employed microchannel heat sink with and without nano-encapsulated phase change material slurry as coolant

The application of hydrofoil-shaped sidewall ribs (SWRs) in microchannel heat sinks (MCHSs) with nano-encapsulated phase change material (NEPCM) particle slurry as coolant has recently been proven as an improvised solution for enhancing heat transfer, with a reasonable penalty in pressure drop. However, the further mitigation of high-power densities of modern micro/nanoscale devices is limited owing to manufacturing complications and design constraints. By using the geometrical and thermophysical parameters of hydrofoil-shaped SWRs and NEPCM slurry, the present study numerically evaluates the thermal and hydrodynamic performance of an MCHS device. The preliminary results reveal the superior performance of NEPCM slurry with hydrofoil-shaped SWRs compared with that of water without ribs in an MCHS owing to the improved latent heat storage capability of the former fluid as well as the enhanced heat transfer area of the rib structure. Increase in the width and height of hydrofoil-shaped SWRs and particle diameter, latent heat of fusion and mass concentration of the NEPCM slurry at a constant Reynolds number (Re) and inlet temperature enhances heat transfer. However, increase in the pitch of the hydrofoil-shaped SWRs and melting temperature range of the NEPCM slurry degrades heat transfer. The inlet temperature of the NEPCM slurry, with a slightly lower value than the melting point of the NEPCM particles, yields superior performance. The maximum ratios of the Nusselt number, that is, Nuavg,SWR/Nuavg,ref and Nuavg,NEPCM/Nuavg,ref, are 1.88 and 2.68, respectively. A particle loading of more than 10% at a high Re of 1000 is not reasonable from the thermodynamic efficiency perspective. The findings are expected to serve as a benchmark for the selection of the geometric, thermophysical and operational parameters of future MCHS devices with hydrofoil-shaped SWRs and NEPCM slurry as coolant.

Effects of Inert Fuel Diluents on Thermoacoustic Instabilities in Gas Turbine Combustion

The effects of inert diluents in the fuel mixture of a model swirl stabilized gas turbine combustor, under thermoacoustically unstable limit cycle operation, were studied experimentally. The measurements included particle image velocimetry (PIV), high-speed CH* chemiluminescent imaging, and dynamic pressure signals. The paper focuses on the dynamic phenomenon of the period doubling bifurcation, which came about when the equivalence ratio of undiluted flames was enriched from 0.55 to 0.60 under constant Reynolds number (). The bifurcation featured an emergence of an aerodynamically related timescale in addition to the fundamental timescale that was induced from an unstable acoustic eigenmode. The aerodynamic timescale is introduced by azimuthal convection of a high-heat-release-rate region and is linked in the literature with a precessing motion of the recirculation zone. It was found that on increasing the diluent molar fraction up to 50%, the amplitude and the frequency of the limit cycle fundamental acoustic mode decreased. A mechanism to interpret this suppression is that increasing the diluent molar fraction of the fuel makes the flame more susceptible to quenching because the extinction strain rate of the mixture is decreased. The paper argues that the existence of inert diluents in the fuel can significantly alter the dynamic state of the combustor, because the anchoring locations of the flame greatly depend on the composition-sensitive extinction strain rate of the mixture.

Investigation on flow and heat transfer of compact brazed plate heat exchanger with lung pattern

This study relates to an evaluation of the thermo-hydraulic performance for a state-of-the-art compact brazer plate heat exchanger (PHE) in 3 kW for 3-plates. To improve heat transfer and pressure drop of the PHE, a lung pattern is designed at certain heights on the plate surface by using biomimetic approach in first time. It is known that human pulmonary system is one of the most effective heat exchanger devices in the nature. Its tree like structure provides good compactness, consequently better heat transfer rate per unit volume. By means of achievements in additive manufacturing technologies in recent years, a heat exchanger with any desired plate or fin geometry can be manufactured. In this study, plate surfaces have been designed as a 3D lung like structure in order to increase the heat transfer area. CFD simulations are then performed with Ansys-Fluent program in operating condition under supply temperatures of 90 °C and 40 °C with a mass flow rate of 0.05 kg/s for hot and cold sides, respectively. The simulation is validated by the numerical and experimental results of an existing Chevron type compact brazed PHE. The results show that there is a 71.30% increase in heat transfer and a 67.8% decrease in pressure drop for 6.66% less volume compared to the reference PHE. The effectiveness of the lung patterned PHE is found to be 0.350. Fluid velocity in plate cavities on lung pattern is quite irregular due to turbulence formation at almost constant Reynolds numbers. This increases its effectiveness. Finally, lung patterned plates designed with biomimetric method is a guideline study to improve the performance of PHEs.

A particle image velocimetry study of dual-rotor counter-rotating wind turbine near wake

This experimental work studied the flow characteristics in the near wake region behind dual-rotor wind turbines using two-dimensional particle image velocimetry. Two auxiliary rotors of 50% and 80% scale of the main rotor were installed upwind and operated in counter-rotating condition, which are compared to the conventional single-rotor turbine. In all the three configurations, a constant Reynolds number 9.5times 10^4 was applied, and all the rotors operated at a fixed tip speed ratio of 3.46. The mean and phase-averaged velocity fields were investigated together with the turbulence kinetic energy. It was found that the two auxiliary rotors do not result in a significantly different wake flow property. The configuration implementing the 50% auxiliary rotor sees a slightly better wake characteristics, in terms of weaker main rotor tip vortices and a counter-rotating swirling shear region at the mid-span behind the main rotor. The decay rates of the peak vorticity of the main rotor tip vortices and their circulation are found to follow an exponential manner.Graphic abstract

A numerical study on influence of the control cylinder on two side-by-side cylinders

Flow past two cylinders is a complex problem especially when the cylinders are close together. Due to the strong coupling between the generated vortices of both cylinders, the intricate dynamics of the system is significantly altered when compared to flow past a single circular cylinder and when two cylinders are further apart. In this study, the effect of the introduction of a control cylinder is observed along with a two cylinder system. Though this topic is challenging, it has rarely been previously addressed. The control cylinder has a diameter $$\frac{1}{7}\hbox {th}$$ of the main cylinder diameter (D). The position of the control cylinder has been varied downstream along the x-axis. A constant Reynolds Number ($$\hbox {Re} = 100$$) is used for all numerical simulations. To understand the flow physics, tools such as proper orthogonal decomposition, $$\lambda _2$$ criterion, and fast Fourier transforms have been employed. It is seen that when the control cylinder is moved downstream, vortices from the two cylinders are more periodic. But, beyond a certain distance, moving the control cylinder further downstream, the influence of the control cylinder is lost and the flow becomes aperiodic.

사각 채널 내의 볼텍스 유동에 관한 실험적 연구(2)

Mixed convective vortex flow in the 3-D horizontal rectangular channel filled with high viscous fluid is investigated for various aspect ratio experimentally. The particle image velocimetry technique with thermo-sensitive liquid crystal tracers (PIT) is used for visualizing and analysis. This method allows simultaneous measurement of velocity and temperature. Quantitative data of temperature and velocity are obtained by applying the color-image processing to a visualized image, and neural network is applied to the color-to-temperature calibration. In this study, the working fluid is silicon oil (Pr = 909), the aspect ratio (channel width to heigh) is varied from 2 to 6 at constant Reynolds number (Re = 4×10<SUP>-3</SUP>). From the present study, the temperature and velocity of mixed convective vortex flow can be visualized simultaneously, and it is found that the longitudinal roll number in the channel is increased with increasing aspect ratio, and the experimental results agree well with numerical results.

Convective heat transfer studies in dilute alumina and silica nanofluids flowing through a channel using Mach-Zehnder interferometry

Convective heat transfer in dilute oxide nanofluids flowing through a channel was measured using the Mach-Zehnder interferometry, a non-intrusive measurement technique. The convective performance of two different concentrations (0.01 and 0.02 vol.%) of alumina and silica nanofluids were studied and compared with that of de-ionized (D.I) water. In the present work, fringe analysis based on Naylor and Duarte method, with appropriate modifications for liquids was used to measure the local heat transfer coefficients along the channel. Studies have previously been reported only on the effects of Reynolds number in the channel flow, at a single heat input. In the present work, experiments were conducted at a constant Reynolds number (200), for six different heat inputs in the range of 5 W to 30 W. At 30 W, the enhancement in average heat transfer coefficient for 0.02 vol.% alumina nanofluid was 15.9 ± 4.1%, while, 0.02 vol.% silica nanofluid showed a decrement of 11.9 ± 2.9% compared to D.I water. Silica nanofluids were found performing inferior to D.I water and alumina nanofluids for all heat inputs. The variations in the trend for silica nanofluids are explained in the light of thermophoretic effects, Brownian motion and disruption of boundary layer.

Simulation of blood flow in arteries with aneurysm: Lattice Boltzmann Approach (LBM)

Background and ObjectiveIn most countries, the higher death rates are due to cardiovascular disease and stroke. These problems often derive from irregular blood flow and the circulatory system disorder. MethodsIn this paper, the blood flow is simulated in a created aneurysm in the artery upon using Lattice Boltzmann Method (LBM). Blood is selected as a non-Newtonian fluid which was simulated with power-law model. The lattice Boltzmann results for non-Newtonian fluid flow with power-law model and the curved boundary are compared and validated with previous studies which show a good agreement. In this study, simulations are carried out for two types of aneurysms. For the first aneurysm, three power-law exponents of 0.6, 0.8 and 1.0 at Reynolds number of 100 for three different cases are investigated. ResultsThe results show that the wall shear stress increases with increasing the power-law exponent. In addition, in the main duct of artery where the velocity is larger, shear stress is lower due to the smaller velocity gradient. For the second Aneurysm, the simulations are done for three Reynolds numbers of 100, 150 and 200, and three Womersley numbers of 4, 12 and 20. The blood flow is pulsating at the inlet such as the real pulsating wave in the blood. Results show that with increasing the Womersley number, the velocity profiles in the middle of the aneurysm are closer at a constant Reynolds number. ConclusionsWith increasing the Reynolds number, the range of vortices and values of velocity and tension grow in the aneurysm.

Effects of Pr and pool curvature on thermocapillary flow instabilities in annular pool

We linearly analyzed the effect of the Prandtl number (Pr) on the stability of thermocapillary flow in shallow annular pools with an aspect ratio Γ = (Ro - Ri)/d = 20 and two radius ratios ΓR = Ri/Ro= 0.50 and 0.98039, where d, Ro, and Ri are the liquid depth and the radii of the heated outer wall and the cooled inner wall, respectively. The results for Pr ∈ [0, 102] show that the steady axisymmetric thermocapillary flows in these annular pools become unstable against oscillatory instability modes, OSC1, OSC2 and hydrothermal wave (HTW). Two co-dimension-two bifurcations occur at Pr*1 and Pr*2. The critical mode for Pr ≤ Pr*1 is OSC2 with almost constant critical Reynolds number Rec, large wave number and high frequency. OSC1 with smaller wave number and lower frequency is the critical mode for Pr*1 ≤ Pr ≤ Pr*2 and Rec slightly decreases with increasing Pr. HTW is the critical mode for Pr ≥ Pr*2 and Rec decreases with increasing Pr. The values of (Pr*1, Pr*2) are (0.00953, 0.03054) for ΓR = 0.50, and (0.01919, 0.01946) for ΓR = 0.98039. Energy-budget analyses reveal that the instabilities in low-Pr range are caused by instabilities in the steady toroidal vortex (or vortices) near the cold wall.

Numerical Study of Laminar Flow Separation in Step Geometry Using Corners Formulation in Lattice Boltzmann Method

In this study, the velocity and temperature distributions of incompressible laminar flow of a step downstream have been investigated using the lattice Boltzmann method. The inlet and outlet flows at internal and external corners of this step have been simulated using the single relaxation time model in various Reynolds number and expansions ratios, two dimensionally. According to Zou and He, constant temperature boundary conditions and open-boundary condition have been utilized at inlet and outlet, respectively. The model results have been validated by experimental data. It was obtained that there are agreements between them. It was resulted that the length of recirculation zone is a function of Reynolds number of inlet flow. Increasing the Prandtl number decays the rate of fluid temperature decrement. Furthermore, in constant Reynolds and Prandtl numbers, decreasing the expansion ratio decreases the Nusselt number of flow. Generally, the ability of lattice Boltzmann method in simulating complex fluid and thermal flows has been shown.