Heat Transfer Through a Compressible Laminar Boundary Layer

1962 ◽  
Vol 13 (3) ◽  
pp. 255-270 ◽  
Author(s):  
N. Curle
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Pressure Gradient ◽  
Heat Transfer Rate ◽  
Laminar Boundary Layer ◽  
Wall Temperature ◽  
Transfer Rate ◽  
Boundary Layer Separation ◽  
Accurate Solution ◽  
Simple Method

SummaryA simple method is presented for calculating the heat transfer rate through a compressible laminar boundary layer. The temperature of the wall is assumed to be uniform, the viscosity being proportional to the absolute temperature with a ratio which may vary in an appropriate manner with position along the wall; the Prandtl number is arbitrary but greater than about 0·5.The method assumes, following Lighthill, that heat transfer rates are determined mainly by the form of the velocity field in regions close to the wall. After applying the Ulingworth-Stewartson transformation the velocity is expanded in powers of the distance Y normal to the wall, three terms being retained. The first term, proportional to the skin friction, is zero near to a position of boundary layer separation. The second term is proportional to the pressure gradient and also to the wall temperature. Accordingly it can become close to zero when the wall is sufficiently cooled. The third term becomes important only when the first two are simultaneously close to zero and is proportional to the heat transfer rate. Since the three terms cannot, in fact, ever be all zero at the same position they form a uniformly valid non-trivial approximation to the velocity close to the wall.With this velocity profile, and following similarity arguments first given by Liepmann, it is possible to reduce the integrated thermal energy equation to a first-order ordinary differential equation for the heat transfer rate, which is easily solved.A comparison with an accurate solution of Poots, for a particular pressure gradient and wall temperature, yields agreement to within 1 per cent at one third of the distance from leading edge to separation and 3 per cent at twice this distance, with the error rising to 16 per cent at separation where the heat transfer rate is, of course, very low. The modesty of this error is very encouraging.

Contributions to the theory of heat transfer through a laminar boundary layer

1950 ◽  
Vol 202 (1070) ◽  
pp. 359-377 ◽  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Temperature Distribution ◽  
Skin Friction ◽  
Heat Transfer Rate ◽  
Wall Temperature ◽  
Transfer Rate ◽  
Arbitrary Distribution ◽  
Main Stream

An approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises’s form, and approximating the coefficients in a manner which is most closely correct near the surface. The heat transfer rate to a portion of surface of length l (measured downstream from the start of the boundary layer) and unit breadth is given as -½ k /(⅓)! (3σρ/μ 2 ) ⅓ ∫ l 0 (∫ l x √{ T ( z )} dz ) ⅔ dT 0 ( x ), where k is the thermal conductivity of the fluid, σ its Prandtl number, ρ its density, μ its viscosity, T ( x ) is the skin friction, and T 0 ( x ) the excess of wall temperature over main stream temperature. A critical appraisement of the formula (§3) indicates that it should be very accurate for large σ, but that for σ of order 0.7 (i. e. for most gases) the constant ½3 ⅓ /(⅓) ! = 0.807 should be replaced by 0.73, when the error should not exceed 8 % for the laminar layers that occur in practical aerodynamics. This yields a formula Nu = 0.52σ ⅓ ( R √ C f ) ⅔ for Nusselt number in terms of the Reynolds number R and the mean square root of the skin friction coefficient C f , in the case of uniform wall temperature. However, for the boundary layer with uniform main stream, the original formula is accurate to within 3% even for σ = 0.7. By known transformations an expression is deduced for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary Mach number (equation (42)). From this, the temperature distribution along such a surface is deduced (§ 4) in the case (of importance at high Mach numbers) when heat transfer to it is balanced entirely by radiation from it. This calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance. This effect will be considerably mitigated for bodies of high thermal conductivity; the author is not in a position to say whether or not it will be appreciable for metal projectiles. But for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence. An appendix shows how the method for approximating and solving von Mises’s equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods.

Heat Transfer in a Laminar Boundary Layer with Constant Fluid Properties and Constant Wall Temperature

1958 ◽  
Vol 62 (565) ◽  
pp. 60-64 ◽  
Author(s):  
A. G. Smith ◽  
D. B. Spalding
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Laminar Flow ◽  
Laminar Boundary Layer ◽  
Wall Temperature ◽  
Two Dimensional ◽  
Constant Wall Temperature ◽  
Simple Method ◽  
Flow Surface ◽  
Fluid Properties

A simple method is given for the calculation of the heat transfer from a laminar flow surface. Computation is by a quadrature. The method is essentially a simplification and extension of the Eckert(1) method, and is applicable both to two–dimensional and to axisymmetric flows.

Non-axisymmetric Homann stagnation point flow and heat transfer past a stretching/shrinking sheet using hybrid nanofluid

2020 ◽  
Vol 30 (10) ◽  
pp. 4583-4606 ◽  
Author(s):  
Najiyah Safwa Khashi’ie ◽  
Norihan Md Arifin ◽  
Ioan Pop ◽  
Roslinda Nazar ◽  
Ezad Hafidz Hafidzuddin ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Strain Rate ◽  
Stagnation Point ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Hybrid Nanofluid ◽  
Ambient Fluid ◽  
Content Type ◽  
Flow And Heat Transfer

Purpose This paper aims to scrutinize the analysis of non-axisymmetric Homann stagnation point flow and heat transfer of hybrid Cu-Al2O3/water nanofluid over a stretching/shrinking flat plate. Design/methodology/approach The similarity transformation which fulfils the continuity equation is opted to transform the coupled momentum and energy equations into the nonlinear ordinary differential equations. Numerical solutions which are elucidated in the tables and graphs are obtained using the bvp4c solver. Findings Non-unique solutions (first and second) are feasible for both stretching and shrinking cases within the specific values of the parameters. First solution is the physical/real solution based on the execution of stability analysis. An upsurge of the ratio of the ambient fluid strain rate to the plate strain rate can delay the boundary layer separation, whereas a boost of the ratio of the ambient fluid shear rate to the plate strain rate only accelerates the separation of boundary layer. The heat transfer rate of hybrid nanofluid is greater for the stretching case than the shrinking case. However, for the shrinking case, the heat transfer rate intensifies with the increment of the copper (Cu) nanoparticles volume fraction, whereas a contrary result is found for the stretching case. Originality/value The present numerical results are original and new. It can contribute to other researchers on electing the relevant parameters to optimize the heat transfer process in the modern industry, and the right parameters to generate non-unique solution so that no misjudgment on flow and heat transfer features.

Heat Transfer on a Flat Surface Under a Region of Turbulent Separation

1992 ◽  
Author(s):  
R. B. Rivir ◽  
J. P. Johnston ◽  
J. K. Eaton
Keyword(s):  
Heat Transfer ◽  
Pressure Gradient ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Separation Bubble ◽  
Adverse Pressure Gradient ◽  
Constant Heat Flux ◽  
Opposite Wall ◽  
Turbulent Separation ◽  
Full Separation

Fluid dynamics and heat transfer measurements were performed for a separation bubble formed on a smooth, flat, constant-heat-flux plate. The separation was induced by an adverse pressure gradient created by deflection of the opposite wall of the wind tunnel. The heat transfer rate was found to decline monotonically approaching the separation point and reach a broad minimum approximately 60% below zero-pressure-gradient levels. The heat transfer rate increased rapidly approaching reattachment with a peak occuring slightly downstream of the mean reattachment point. The opposite wall shape was varied to reduce the applied adverse pressure gradient. The heat transfer results were similar as long as the pressure gradient was sufficient to cause full separation of the boundary layer.

A Numerical Study of Impulsively Started External Convection at Microscale

2014 ◽  
Vol 31 (1) ◽  
pp. 79-90
Author(s):  
K. Ramadan
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Thermal Boundary Layer ◽  
Numerical Study ◽  
Velocity Slip ◽  
Transition Time ◽  
Surface Cooling ◽  
Linear Ordinary Differential Equations

ABSTRACTImpulsively started external convection at microscale level is studied numerically in both planar and axisymmetric geometries. Using similarity transformation, the resulting coupled partial and non-linear ordinary differential equations are simultaneously solved by finite differences together with a well established ordinary differential equation solver, over a range of problem parameters. Rarefaction effects within the slip flow regime on the thermal boundary layer response, heat transfer rate and transition time when system experiences sudden changes in surface temperature are analyzed, and a comparison between sudden surface cooling and heating is presented. The results show that the thermal boundary layer thickness, heat transfer rate and the transition time is considerably influenced by the degree of rarefaction. The transition time tends to be less sensitive with increasing rarefaction. The velocity slip and temperature jump factors are found to have opposite effects on the transition time and the heat transfer rate, with the velocity slip factor having the most profound influence on these parameters.

The Effects of Small Changes of Prandtl Number and the Viscosity-Temperature Index on Skin Friction

1964 ◽  
Vol 15 (4) ◽  
pp. 392-406 ◽  
Author(s):  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Wall Temperature ◽  
External Pressure ◽  
Main Stream ◽  
Pressure Gradients ◽  
Temperature Index

SummaryThe analytic simplifications in boundary-layer analysis that result from the assumptions that the Prandtl number σ and the viscosity-temperature index ω are unity make it desirable to be able to assess the effects of the departures of the actual values of these parameters from unity. In this paper only the effects on skin friction are considered. Formulae of acceptable validity and wide application are first used to produce generalised curves for these effects for given main-stream Mach numbers and wall temperature conditions for the case of zero external pressure gradient for both laminar and turbulent boundary layers (Figs. 1 and 2).A number of calculated results for the laminar boundary layer with favourable and adverse pressure gradients is then analysed (Figs. 3, 4 and 5) and it is shown that these results are consistent with the assumption that, for a given wall temperature, the effects of small changes of σ and ω on skin friction are independent of the external gradient, so that the appropriate curves of Figs. 1 and 2 apply. Where the change of a- is associated with a change of wall temperature (e.g. if the heat transfer is specified as zero) then the interaction between pressure gradient and this temperature change can be significant in its effects on skin friction for the laminar boundary layer and can only be assessed if the effects of changes of wall temperature with constant σ and ω have been separately determined for the pressure distribution considered. It is inferred that in all cases, except with large adverse pressure gradients and imminent separation, the effects of changes of ω and σ for the turbulent boundary layer are reliably predicted by the zero pressure gradient curves of Figs. 1 and 2 and the effect of any associated change of wall temperature can then be reliably inferred from the zero pressure gradient formula (equation (15)) in the absence of more specific calculations covering a range of wall temperatures.

Heat Transfer Enhancement of Channel Flow via Vortex-Induced Vibration of Flexible Cylinder

2014 ◽  
Author(s):  
Junxiang Shi ◽  
Jingwen Hu ◽  
Steven R. Schafer ◽  
Chung-Lung (C. L. ) Chen
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Convective Heat Transfer ◽  
Heat Transfer Rate ◽  
Channel Flow ◽  
Convective Heat ◽  
Thermal Performance ◽  
Transfer Rate ◽  
Thermal Boundary Layer ◽  
Flexible Cylinder

Thermal diffusion in a developed thermal boundary layer is considered as an obstacle for improving the forced convective heat transfer rate of a channel flow. In this work, a novel, self-agitating method that takes advantage of vortex-induced vibration (VIV) is introduced to disrupt the thermal boundary layer and thereby enhance the thermal performance. A flexible cylinder is placed at the centerline of a rectangular channel. The vortex shedding due to the cylinder gives rise to a periodic vibration of the cylinder. Consequently, the flow-structure-interaction (FSI) strengthens the disruption of the thermal boundary layer by vortex interaction with the walls, and improves the mixing process. This new concept for enhancing the convective heat transfer rate is demonstrated by a three-dimensional modeling study at different Reynolds numbers (84∼168). The fluid dynamics and thermal performance are analyzed in terms of vortex dynamics, temperature fields, local and average Nusselt numbers, and pressure loss. The channel with the self-agitated cylinder is verified to significantly increase the convective heat transfer coefficient. When the Reynolds number is 168, the channel with the VIV improves the average Nu by 234.8% and 51.4% as opposed to the clean channel and the channel with a stationary cylinder, respectively.

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