The Influence of Prandtl Number on Shear Stress and Heat Transfer in Compressible Boundary Layers

 You're right, in compressible boundary layers, increasing the Prandtl number (Pr) generally leads to a decrease in shear stress (skin friction) and Stanton number. Here's why:

Prandtl Number (Pr):

Pr is the ratio of momentum diffusivity (ν) to thermal diffusivity (α). It represents the relative ease with which momentum and heat diffuse within a fluid.

• Low Pr: When Pr is low (typical for gases), momentum diffuses more readily than heat. This means velocity gradients (shear) are higher near the wall, resulting in higher skin friction.
• High Pr: As Pr increases (typical for liquids), thermal diffusivity becomes more dominant. Heat diffuses faster, reducing the temperature gradients near the wall. This, in turn, affects the viscosity (which is temperature-dependent) and reduces the shear stress at the wall.

Impact on Shear Stress and Skin Friction:

• Higher Pr: With reduced temperature gradients at the wall due to higher Pr, the viscosity near the wall decreases. This weaker viscous resistance leads to a decrease in shear stress and consequently, a decrease in skin friction coefficient.

Impact on Stanton Number:

• Stanton Number (St): This non-dimensional number represents the ratio of convective heat transfer to conductive heat transfer at the wall.
• Higher Pr: As mentioned earlier, higher Pr reduces the temperature gradient near the wall. This weaker temperature gradient also leads to a decrease in the convective heat transfer rate. Since St is directly proportional to the convective heat transfer rate, it also decreases with increasing Pr.

Additional Considerations:

• The relationship between Pr and shear stress/skin friction is more pronounced at lower Mach numbers (incompressible limit). At higher Mach numbers, compressibility effects can play a more significant role.
• While the general trend is a decrease, the specific impact of Pr on these parameters depends on the specific flow conditions and boundary layer characteristics.

In summary, increasing Prandtl number in a compressible boundary layer leads to a decrease in shear stress (skin friction) and Stanton number due to reduced temperature gradients near the wall and the resulting changes in viscosity and convective heat transfer rates.