Aerodynamic Heating

Mech 448 QUEEN'S UNIVERSITY Faculty of Applied Science Department of Mechanical and Materials Engineering

AERODYNAMIC HEATING

MECH 448 September, 2010

COMPRESSIBLE FLUID FLOW

Mech 448

Mech 448

INTRODUCTION When a gas flows over a surface, the gas in contact with the surface is brought to rest as a result of viscosity. Associated with this decrease in velocity at a surface is a rise in temperature. If the gas velocity is high, this temperature rise associated with the slowing of the flow near a surface can become quite large. This, basically, is what is referred to as the "aerodynamic heating" of a surface. The phenomena is illustrated in following figure.

The velocity is zero at the surface because of the action of viscosity and the temperature rise in the boundary layer is the result of the work done on the flow by the viscous forces, i.e. the temperature rise is produced by the dissipation of kinetic energy into heat as the result of the work done by the viscous forces. The temperature rise, i.e. the aerodynamic heating, is, therefore, said to be the result of "viscous dissipation".

Temperature Rise Near the Surface of a Body.

Mech 448

Aerodynamic heating is particularly important in hypersonic flows. However in such flows changes in the chemical nature of the gas can occur due to the very high temperatures existing in such flows. Further, because the temperature rises at the surface are so high in hypersonic flow, radiation heat transfer can become important. Attention will, therefore, here be restricted to supersonic flows. Radiation effects will be ignored in this chapter.

Mech 448 THE ADIABATIC SURFACE TEMPERATURE Consider flow over a nearly flat surface at a Mach number M as shown in following figure.

Flow Situation Considered

Mech 448

Mech 448

It is assumed that the surface is adiabatic, i.e. there is no heat transfer to or from the plate. In this case, the surface temperature is denoted by Twad. Consider two points A and B as shown in the figure. Point A is in the freestream outside the boundary layer and point B is on the surface. If the flow between points A and B is adiabatic, then, since M at point B on the surface of the plate is zero, it follows as shown before that:

TWad T∞

= 1+

γ −1 2

M 12 =

T0 T∞

where To is the stagnation temperature.

Heat Transfer in Boundary Layer Away From the Surface

Mech 448

Mech 448 For this reason it is usual to define a recovery factor, r, such that:

However, the actual process between A and B is not adiabatic. This is because as the temperature rises as the plate is approached there is a heat transfer towards the colder gas in the freestream. This is shown in the following figure.

TWad − T∞ T0 − T∞

The value of r, for a given geometrical flow situation, according to dimensionless analysis is a function of the Reynolds number, the Mach number and the Prandtl number i.e.:

=r

r is a measure of the fraction of the local freestream dynamic temperature rise that is actually "recovered" at the wall. It is defined as the ratio of the actual rise in the temperature of the gas across the boundary layer to the maximum possible rise in temperature that could occur.

Mech 448

r = f ( Re, M , Pr ) The Prandtl number is a property of the fluid involved being equal to approximately 0.7 for air. Experimental and analytical studies indicate that for flow over a near flat surface, the Reynolds number only effects the value of the recovery factor by determining whether the flow in the boundary layer is laminar or turbulent and that the Mach number has a negligible effect on the recovery factor. This is illustrated by the results shown in the following two figures.

Mech 448 Hence, experimental and numerical studies indicate that: Typical Effect of Mach Number on the Recovery Factor For a Laminar Boundary Layer

r = f ( Pr ) the function being different in laminar and turbulent boundary flow. Experimental and analytical studies indicate that:

For laminar Flow : r = Pr1/ 2 Typical Effect of Reynolds Number on the Recovery Factor

For turbulent Flow : r = Pr1/ 3

Mech 448

Mech 448

Now, it will be noted from the equation defining the recovery factor that:

r=

Twad / T∞ − 1 T0 / T∞ − 1

Consider air flowing at a temperature of -40oC flowing over an adiabatic flat plate. Assuming turbulent boundary layer flow, the plate temperature variation with Mach number is as shown in the graph.

but as mentioned above:

T0 ⎛ γ −1⎞ 2 = 1+ ⎜ ⎟M T∞ ⎝ 2 ⎠

Variation of adiabatic plate temperature with flow Mach number.

Hence:

Twad / T∞ − 1 Twad ⎛ γ −1⎞ 2 i.e.: = 1+ r ⎜ ⎛ γ −1⎞ 2 ⎟M M T ⎜ ⎟ ⎝ 2 ⎠ ∞ ⎝ 2 ⎠ This allows the adiabatic surface temperature for a near flat surface to be determined. r=

Mech 448

Mech 448

Concorde

In low speed flow, i.e. M 1 , if the surface is not adiabatic but is heated to a temperature, Tw, the magnitude and direction of the heat transfer from the surface will depend on the difference between the wall temperature and the fluid temperature. The rate of heat transfer is in such a case, therefore, expressed as:

Q = hA(TW − T f )

SR 71 BlackBird

where Tf is the fluid temperature, A is the surface area and h is the heat transfer coefficient. The fluid temperature Tf is usually taken as the temperature in the freestream outside the boundary layer.

Mech 448

Mech 448

In high speed flow, however, it is to be expected that the magnitude and direction of the heat transfer at the surface will depend on the difference between the wall temperature and the adiabatic wall temperature i.e. if Tw is less than Twad there will be heat transfer from the fluid to the surface while if if Tw is greater than Twad there will be heat transfer from surface to the fluid. Hence it is usual in high speed gas flows to write:

Q = hA(TW − TWad )

HEAT TRANSFER AND SHEAR STRESS IN HIGH SPEED FLOW: It was noted above that in high speed flow the heat transfer rate is expressed as:

Q = hA(TW − TWad ) Thus in order to find the heat transfer rate, the value of h, the heat transfer coefficient, has to be found. Now, equations for the heat transfer coefficient are usually expressed in dimensionless form. For this purpose, the Nusselt number, Nu, is introduced, this being defined by:

Nu =

hL k

Here, L is some appropriate measure of the size of body being considered. For example, in the case of a wide flat plate, L would be the length of the plate in the flow direction.

Mech 448

Mech 448 Now, in low speed flow, i.e., flow in which viscous dissipation effects are negligible:

Nu = function ( Re, Pr ) The fluid properties in the dimensionless numbers, e.g. the thermal conductivity k , are evaluated at the average of the surface and the fluid temperatures. In high speed gas flows it would be expected that:

Nu = function ( Re, Pr, M ) However, it has been found using both experimental and theoretical results that provided the gas properties are evaluated at a suitable mean temperature, the direct effect of the Mach number can be neglected.

The Mach number will have an indirect effect because it will, in general, influence the mean temperature used to find the gas properties. It has further been found that provided the correct mean temperature is used to find the gas properties, the same equation that gives the Nusselt number in low speed flow can be used in high speed flow. For example, consider boundary layer flow over a flat plate. Experimental and theoretical studies have indicated that the following equations for the Nusselt number applies in low speed flow: Laminar boundary layer:

Nu L = 0.664 Re1/L 2 Pr1/ 3

Turbulent boundary layer:

Nu L = 0.037 ReL0.8 Pr1 / 3

Mech 448

Mech 448 In these equations NuL is the Nusselt number based on L , i.e., h L/ k , and ReL is the Reynolds number also based on the length of the plate L. Transition from laminar to turbulent flow occurs approximately at ReL = 106. Theoretical and experimental results have shown that these equations can be used in high speed flow provided the fluid properties are evaluated at:

OTHER GEOMETRICAL SITUATIONS: The discussion of heat transfer in high speed flow given above was concerned with flow over a flat or near flat surface. The equations given will give good results, for example, for the flow over the blades in turbomachines outside the area of the stagnation point and for flow over the wings and fuselage of an aircraft outside the area of the stagnation point. This is illustrated in the following figure.

Tprop = T1 + 0.5 (Tw - T1 )+ 0.22 (Twad - T1 ) In low speed flow where Twad = T1 the above equation gives:

Applicability of Flat Plate Equations

Tprop = T1 + 0.5 (Tw - T1 ) Thus in low speed flow the properties are evaluated at the average of the wall and fluid temperatures.

Mech 448

Mech 448

In other situations, related approaches can be used but the value of the recovery factor may be different from that which applies to flat plate flows and the equation for the Nusselt number will normally be different from that for flat plate flow. For example, consider the heat transfer rate near the stagnation point of a body placed in a supersonic gas flow. This situation is shown in following figure.

Stagnation Point Region in Supersonic Flow.

As shown in the above figure an effectively normal shock forms ahead of the stagnation point. The flow through this shock is adiabatic and the flow downstream of this shock is subsonic. The flow outside the boundary layer in the stagnation point region will, therefore, be near the stagnation temperature and the increase in temperature across this boundary layer due to viscous dissipation will be relatively small and so the heat transfer rate across the boundary layer resulting from this temperature increase will be small. This means that if the wall in the stagnation point region is adiabatic, it will effectively be at the stagnation temperature. Hence, for flow in the stagnation point region, the recovery factor is effectively equal to 1.

Mech 448

Mech 448

If the wall is not adiabatic but is kept at a temperature Tw, this means that the heat transfer rate in the stagnation region per unit area should be written as:

q = h(Tw − T0 ) The value of h can be obtained by using the equations for low speed flow near a stagnation point and, because the flow outside the boundary layer is subsonic, evaluating the gas properties at the average of the wall and the temperature behind the normal shock wave. If the boundary layer near the stagnation point is two-dimensional and can be assumed to be laminar, the heat transfer rate in this region is approximately given by:

Nu D = 1.14 ReD0.5 Pr 0.4

Mech 448

CONCLUDING REMARKS As a result of viscosity, the velocity at the surface over which a gas is flowing is zero. Therefore, the gas velocity decreases as the surface is approached. As a consequence, the gas temperature rises causing so-called aerodynamic heating of the surface. An expression has been given which allows the temperature of an adiabatic surface to be found, this expression involving the recovery factor. When the surface is not adiabatic, the heat transfer rate from the surface depends on the difference between the surface temperature and the adiabatic surface temperature.

Here NuD and ReD are the Nusselt and Reynolds numbers based on the effective diameter of the body in the stagnation point region as shown in the following figure. The Reynolds number is based on the velocity behind the shock wave.

Stagnation Point Diameter

The approach outlined above is very approximate but should serve to illustrate some of the main features involved in the analysis of stagnation point heat transfer.