[Behemont]

Hi, I’m simulating a compressible boundary layer, I would like to understand why the shear stress or the skin friction and the Stanton number decreases as Prandtl number increases.
Thank you

[Martin_Sz]

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.**

[Behemont]

Thank you very much, this is really helpful.
Please, can you explain this:
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.