The time-averaged differential equation for energy in a given flow field is linear in the temperature if fluid properties are considered to be independent of temperature. Thus, the concept of a Heat Transfer Coefficient arises such that the heat transfer rate from a wall is given by:
where the heat transfer coefficient, α, is only a function of the flow field. Tw is the wall temperature and Tr, the recovery or adiabatic wall temperature. The above is also true of the Boundary Layer energy equation, which is a particular case of the general energy equation. When fluids encounter solid boundaries, the fluid in contact with the wall is at rest and viscous effects thus retard a layer in the vicinity of the wall. For large Reynolds Numbers based on distance from the leading edge, these viscous layers are thin compared to this length.
When the wall is at a different temperature to the fluid, there is similarly a small region where the temperature varies. These regions are the velocity and thermal boundary layers. In 1905 Prandtl showed that this thin region could be analyzed separately from the bulk fluid flow in that pressure variation normal to the wall may be neglected and the pressure is given by that impressed by the free stream. Velocity normal to the wall is also of order, of the thickness of the boundary layer, the characteristic velocity being that of the free stream and the length being the distance from the leading edge. Thus, the boundary layer equations for steady incompressible laminar flow in two dimensions may be approximated to be:
p, T, u and v are the flow pressure, temperature and velocities along and perpendicular to the surface, respectively. λ, μ, cp and ρ are similarly the thermal conductivity, viscosity, specific heat and density. x and y are Cartesian coordinates along and perpendicular to the surface.
The classical laminar solution to the momentum equation was provided by Blasius for the case of a semi-infinite flat plate aligned with uniform flow. The velocities normalized by the free-stream value u0 are plotted in Figure 1 vs. the nondimensional quantity η = y/xRex−1/2 Rex is the Reynolds number based on distance from the leading edge of the plate.
Figure 1. Laminar boundary layer normalized velocities along and perpendicular to a flat plate from Young (1989).
The velocity gradient at the wall gives the skin friction, τ, which can be expressed as the skin friction coefficient:
The suffix o refers to the free-stream value.
The energy equation may be solved using the Blasius solution to give the heat transfer in terms of the Nusselt Number, Nux, when the dissipation term, μ(∂u/∂y)2 is neglected.
This may also be expressed in terms of a Stanton Number, St = α/ρucp, as, Re St Pr = Nu. Figure 2 shows temperature profiles for different Prandtl Numbers. There is thus an analogy between heat transfer skin friction (Reynolds analogy) which may be expressed as:
Expressions for the boundary layer deficit thicknesses of mass, momentum and temperature are, respectively:
The deficit thickness represents the height of free-stream fluid which carries the boundary layer deficit in the relevant quantity
By integrating the boundary layer equations with respect to y, the integral boundary layer equations are generated in terms of the boundary layer thicknesses. These form the basis for many approximate solutions [Young (1989), Kays and Crawford (1993)].
Figure 2. Normalized temperature profiles in a laminar boundary layer for different Prandtl numbers from Kays and Crawford (1993).
The analysis of boundary layers on wedges, where the external velocity varies as a power law from the leading edge, has been performed by Falkner and Skan. Approximate methods are available for arbitrary free-stream velocity variations [see Young (1989), Kays and Crawford (1993), and Schlichting (1987)]. (See also Wedge Flows.) The two-dimensional stagnation point may be treated as the special case of a wedge of 90° half angle and gives a constant boundary layer thickness. This result is also given by the Hiemenz solution [Schlichting (1987)]. The stagnation point Nusselt number, based on R for a cylinder of radius R, may thus be evaluated as:
For an axisymmetric stagnation point boundary layer, the result becomes:
The boundary layer analysis described above may be considered as a particular case of general perturbation methods in solving the differential equations of fluid mechanics [Van Dyke (1964)]. Perturbation methods add a mathematical rigor to the boundary layer concept and also allow solutions to be determined for variable property flows. Gersten (1982) made a review of perturbation methods in heat transfer, showing that the effects of variable properties may be taken into account with property ratios raised to a relevant power, i.e.,
Suffix w refers to the wall value.
Herwig et al. (1989) have developed this analysis further. Ziugzda and Zukauskas (1989) used experimental data to derive empirical relationships that take temperature-dependent properties into account for laminar and turbulent boundary layers.
The boundary becomes turbulent in response to external disturbances. This transition to a fully turbulent boundary may take place over a significant length of the surface, and is an important factor in the heat transfer to turbine blades, for example, [Mayle (1991)]. The transition takes place in a characteristic manner by the formation of turbulent spots. These spots convect along the surface and grow, so as to coalesce, to finally form the turbulent boundary layer as shown in Figure 3.
The dynamics of the turbulent spot are presently the subject of much experimental and theoretical study [Smith (1994), Kachanov (1994)]. The spreading of spots is inhibited at high free-stream Mach number and also by accelerating flow (favorable pressure gradients). Adverse pressure gradients, on the other hand, cause very rapid formation of the turbulent boundary layer. An interesting property of turbulent spots is that there appears to be a wake region in which generation of spots is inhibited.
It is an experimental observation that after a short inception stage, the heat transfer to the surface under the spot is closely given by that under a continuous turbulent boundary layer, which has grown from the point where spots are first initiated. A typical variation of heat transfer in the transition region is given in Figure 4. Turbulent spots can be seen in this figure passing over consecutive gauges and growing to coalesce to a turbulent boundary layer.
Figure 4. Unsteady simultaneous heat flux signals due to turbulent spots passing over gauges mounted downstream of one another from Clark (1993).
The fraction of time for which the boundary layer is fully turbulent at a point is called the intermittency, γ, and this follows a universal law on a flat plate.
where η = (x – xt)/λ, xt being the point of transition and l the distance between the 25% and 75% intermittency points. The heat transfer may be estimated by assuming that the local Stanton number is given by that time average of the laminar and turbulent values, StL and StT indicated by the intermittency,
A review of such engineering formulas is given in Fraser et al. (1994).
Figure 6. Momentum thickness Reynolds number at the start. The solid line is experimental and the points are Low Reynolds Number Models, from Seiger (1992).
The start of transition and the length of the transition region are notoriously difficult to predict accurately. Classical instability analysis may be used to find those disturbances which first form in the boundary layer as Tollmien–Schlichting waves [Schlichting (1987)]. However, determining when these grow to become nonlinear and form turbulent spots has not been perfected and empirical methods are based on the predicted growth of these waves to a critical point. At free-stream turbulence levels above 1%, the Tollmien-Schlichting wave instability may possibly not have a controlling effect and, so-called, “bypass” transition takes place. In this case, the transition point is a function of free-stream turbulence as indicated in Figure 6. Predictions may be made in this case using low Reynolds number models of turbulent boundary layers, which can also reproduce a laminar boundary layer. Thus free-stream turbulence is entrained, generating a turbulent boundary layer without recourse to a stability analysis, and predictions are given in Figure 6. Other instabilities occur on concave surfaces and Saric (1994) has reviewed these vortex structures, usually termed Goertler vortices. (See Goertler-Taylor Vortex Flow.)
When the boundary layer becomes fully turbulent, the heat transfer through the boundary layer is dominated by the transport associated with the turbulent eddies. Very close to the wall, however, molecular conduction still prevails as the eddies are inhibited by the wall. The time-averaged boundary layer equations become the incompressible constant property case without dissipation.
The - denotes the time-averaged quantity, whereas the ў denotes the fluctuating value. The physical modeling of the terms and , and additional terms in the fully compressible equations are the subject of much study [Cebeci and Bradshaw (1984), Wilcox (1993), Huang Bradshaw and Croakley (1994)]. The traditional teatment is to express the turbulent transport in terms of an eddy viscosity, μI , and mixing length, l, such that
M is the diffusivity and K is a constant.
In the region close to the wall, it can be assumed that the Shear Stress is constant and equal to the wall value, τo. Also the mixing length is taken to be proportional to the distance from the wall. This leads to the law of the wall which may be written in nondimensional terms as:
Similar arguments apply to the temperature field. The region close to the wall is represented by a laminar relationship and that far from the wall, the outer region is characterized by essentially a constant mixing length. Pressure gradients have an influence on the outer region. The mixing length concept may be extended though the viscous sublayer using the Van Driest damping formula for the mixing length, which reduces this to zero exponentially as the wall is approached. Figure 7 shows experimental results confirming the law of the wall. A similar relationship may also be determined for the temperature profiles through the boundary layer, although pressure gradients influence the law of the wall region significantly.
Figure 7. Experimental results compared with the Law of the Wall and the Van Driest sublayer modification, from Kays and Crawford (1993).
From the above analysis of the turbulent boundary layer, the heat transfer to a flat plate in uniform gas flows may be derived.
Boundary layer development is now largely predicted by computing solution to the boundary layer equations with the relevant boundary conditions [Cebeci and Bradshaw (1984), Wilcox (1993)]. More complex turbulence modeling is now routinely used in such codes and subsidiary differential equations for turbulence quantities are solved. For example, the k – model takes M = constant, k2/ , where k is the local turbulence kinetic energy and is the turbulence dissipation rate. Thus, free-stream acceleration and free-stream turbulence may be taken into account. This hierarchy of turbulence models, which allow closure for the governing differential equations, are classified by the number of subsidiary differential equations employed in this closure. Thus the k – model is a two-equation model employing separate differential equations for both k and . Boundary conditions for the turbulence quantities are necessary as the viscous sublayer is approached, and those arising from the law of the wall (i.e., a wall function) may be employed. Alternatively, modifications to the differential equations give rise to low Reynolds number formulations which apply down to the wall and may also represent a laminar boundary layer [Schmidt and Patankar (1991)].
It is possible to model to the term directly without the concept of an eddy viscosity. Reynolds stress transport models make use of the transport equations for the eddy stress and model terms within this equation so as to give the eddy stress term in the boundary layer. A brief survey of these methods and modeling of the transition region is given by Singer (1993).
Compressibility effects occurring in high-speed gas flows may be taken into account in a straightforward manner in computational predictions of the boundary layer through the Equation of State The dissipation term now plays a dominant role in the governing energy equation. Analytical procedures usually allow transformation of the boundary layer equations to facilitate solution. The Crocco Transformation is an example of this for a laminar boundary layer and leads to the conclusion for Prandtl numbers close to unity — which is typical of gases — that the incompressible relationship between heat transfer and skin friction may be employed for a flat plate in a uniform stream. The recovery temperature in this case is found from.
The suffix ∞ refers to the free-stream static value. M is the free-stream Mach number.
The relationship between Stanton number, St, and skin friction coefficient, cf, still applies.
This is independent of Mach number when properties are determined at free-stream conditions. The value of skin friction coefficient is found from the incompressible expression, except that properties are evaluated at a reference temperature.
In turbulent compressible flow, the law of the wall is recovered by a transformation of coordinates proposed by Van Driest. The performance of two-equation turbulence models in the compressible case has been reported by Huang, Bradshaw, and Croakley (1994). Simple corrections to the incompressible heat transfer equations using reference temperatures and temperature ratios may also be employed [Kays and Crawford (1993)].
The boundary layer displacement thickness gives the aerodynamic influence of the boundary layer on the external flow. Usually, this change is small and has little bearing on the growth of the boundary layer itself. At hypersonic speeds, there can be strong effects of displacement as the heating of the boundary layer gas decreases the density and increases this thickness. A parameter on which laminar hypersonic phenomena depend is the hypersonic visous interaction parameter, Weak and strong interactions are recognized. In the former case, there is an effect on the external flow but little on the boundary layer. In the latter, there is a significant influence on both. Anderson (1989) gives an account of this phenomena.
Free-stream turbulence influences the transition from laminar to turbulent flow in the boundary layer, but it also has a very significant effect on the levels of heat transfer within a laminar boundary layer prior to transition. The effect on the stagnation point heat transfer is pronounced and is taken into account using semi-empirical correlations of the form below, where Tu is the free-stream turbulence intensity u'/ū,
Mayle (1994) has generalized this stagnation point enhancement by relating this to the local free-stream acceleration. The effects of fluctuations in free-stream velocity along the surface of a flat plate have been examined by Lighthill (1954) and do not produce such a large increase in surface heat transfer. The influence of free-stream turbulence on turbulent boundary layer heat transfer is not as significant as for a laminar boundary layer. In this case, turbulent intensity and length scale are relevant. Moss and Oldfield (1992) have discussed previous works and produced an empirical correction for the enhancement of heat transfer on a flat plate.
Buoyant flow producing free convection boundary layers is considered in both Cebeci and Bradshaw (1984) and Kays and Crawford (1993). Time-dependent problems are also covered in the texts of Schlichting (1987) and Young (1989). Note that the time to establish a steady laminar boundary layer — when the free-stream is suddenly started — is, of order, the transit time of the free-stream from the leading edge to the point under consideration [Jones et al. (1993)].
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