# Thermal Boundary Layer vs. Hydrodynamic Boundary Layer

1. May 4, 2015

### 0Rion

Hello Guys,

Could someone explain to me the meaning of greater thermal boundary layer over hydrodynamic boundary layer over a flat plate surface? I know how to calculate both streams, but I don't understand the meaning of smaller thermal boundary vs. hydrodynamic boundary and vice-versa. What does the greater thermal boundary layer signify? What does a smaller thermal boundary layer signify?

Thanks!

2. May 4, 2015

### Staff: Mentor

The amount that each boundary layer penetrates into the flow is proportional to the square root of the corresponding diffusivity. For a liquid the thermal diffusivity is typically less than the kinematic viscosity (momentum diffusivity ).

3. May 4, 2015

### HuskyNamedNala

What do you mean by greater xx boundary layer?

The boundary layer can be thought of as a region where the viscos stress dissipates the flow kinetic energy through heat.

The thermal boundary layer is the region where laminar flow convection will dominate and at the smallest scales where conduction dominates.

These boundary layers are coupled. The wall temperature affects viscosity and density, free stream velocity affects boundary layer thickness and associated gradients, viscosity and density affect the heat transfer coefficient, the velocity gradients can actvto increase the flow and wall temperature if high enough. Very complicated and fascinating stuff!

4. May 4, 2015

### HuskyNamedNala

But more generally, you can think of a boundary layer as a region where the tangental transport gradients near the wall are nonzero and decreasing.

5. May 5, 2015

Discussion of viscous dissipation is not really necessary when simply defining the boundary layer in the first place. You only need to point out that the flow "sticks" to the wall and therefore has zero velocity relative to the wall at the wall. Consequently, there is a region near the wall where the velocity decreases from its "inviscid" value to zero and it is only in this region where viscosity is important. That region is the boundary layer. Sure, viscous dissipation occurs, but you don't need it to define the boundary layer in the first place.

Laminar flow has absolutely nothing to do with the definition of the boundary layer in either the thermal or velocity sense. It's really more about the temperature near the surface as compared with the free stream.

For incompressible flows, they are so weakly coupled that they can be (and typically are) treated independently. For compressible flows, there is no way to decouple the two.

I believe you mean wall-normal gradients, as the gradients tangential to the streamlines can certainly still be zero.

If you just want to know what it means in the sense of its definition, it would just mean that it takes a greater distance away from the wall for the temperature to reach the free-stream value than it does for the velocity. If you are looking for physical significance, it could mean a number of things depending on context. It could mean you have a very hot or very cold wall relative to the free stream temperature such that the thermal boundary layer is large compared to the hydrodynamic boundary layer. It could mean that your fluid has a low momentum diffusivity (viscosity) and/or high thermal diffusivity as indicated by a small Prandtl number. It could be a combination of those two things.

6. May 5, 2015

### Staff: Mentor

Yes, and, in many cases, the viscous dissipation has a negligible effect on the velocity and temperature profiles.

Even for incompressible flows, the velocity profile affects the thermal boundary layer but, if the temperature difference between the free stream and the wall is small, the temperature profile does not feed back into the velocity profile. So the temperature profile is coupled to the velocity profile, but the velocity profile is not coupled to the temperature profile.

I don't think that this is the case. As long as the temperature profile does not feed back into the velocity profile, the dimensionless temperature profile within the thermal boundary layer doesn't depend on the temperature difference between the free stream and the wall.

This is the case for liquid metals. But, if the velocity boundary layer is thicker than the thermal boundary layer, then the Prandtl number is high. This is the case with ordinary liquids.

Chet

7. May 5, 2015

### HuskyNamedNala

1) I think it is important to state that losses due to heat contribute to the form of the boundary layer. Otherwise, where does the kinetic energy go? At high speeds (even subsonic Mach numbers based on my experience with a wind tunnel at GE) surface heating has a noticeable impact on engineering systems.

2) I know laminar flow does not strictly come into the definition of the boundary layer. The fact of the matter is that very near the wall the flow must be laminar and that is why the temperature profile takes the basic concave up shape; the rate of heat transfer is proportional to the velocity and flow regime. Without complicating the answer to the question with a turbulent boundary layer or more esoteric cases, I feel the answer I gave suffices.

3) I agree, but in reality they are coupled. Incompressible flow is merely an assumption used to make calculations easier. I think sometimes taking these assumptions for reality gives a false intuition for newcomers.

4) Yes, it was late when I wrote that post!

8. May 5, 2015

### HuskyNamedNala

Is this true even for high temperature gradients? For example T_inf = 298K but T_wall = 400K?

9. May 5, 2015

### Staff: Mentor

The temperature difference has to be small enough for the viscosity to not be affected much (so that the velocity profile is nearly independent of the temperature profile). For this example, with a 100 C temperature difference, in my judgement there would be some effect, although not major.

Chet

10. May 5, 2015

Fair enough. I should be more careful with my language.

You may be correct here. For a zero-pressure-gradient flat plate, that holds true. The ratio of the two thicknesses for the Blasius boundary layer is approximately $\delta_u/\delta_T \approx Pr^{0.4}$. I am not 100% sure this holds true for all incompressible flows, but I am at least leaning your direction now.

The OP was specifically asking about the thermal boundary layer being larger than the velocity boundary layer, thus the reason I limited discussion to small $Pr$.

The actual heat generated by viscous dissipation is typically quite small unless the flow velocity is high and/or the boundary layer is turbulent. The dissipation term in the energy equation is $O(U^2)$, while the conduction terms are $O(c_p\Delta T)$. The latter is usually much higher and dissipation is negligible. This idea is encapsulated in the dimensionless Brinkman number. See Viscous Flow by White for a good reference source on that sort of thing.

I disagree. You can define the thermal boundary layer without any sort of discussion of laminar or turbulent flow. It's simply a matter of the temperature profile. Now, there is certainly a discussion that can be had about how much convection and conduction contribute to the shape of the curve as a function of the wall-normal coordinate and how laminar versus turbulent flow affect that answer, but none of that discussion is necessary to simply define the thermal boundary layer. You also have to be careful about how you describe "the smallest scales," as to many people, that kind of wording carries a certain turbulence-related baggage (i.e. Kolmogorov scales).

I think @Chestermiller clarified my answer quite thoroughly in that the two boundary layers are only decoupled in one direction, but you are correct that it is just an approximation. The problem is, though, that the approximation for low speeds so so nearly exact that you get very small error using it and it is very useful for gaining insight into an otherwise very complex physical situation.

This dissipation function is not a function of temperature and depends only on viscosity and the velocity gradients, so yes, his statement is true so long as the viscosity can reasonable be assumed to be constant. For air, viscosity scales roughly with $\sqrt{T}$ (e.g. Sutherland's law), so while the viscosity will change, it is not usually a very noticeable effect and is often ignored (though it does still exist as a factor). For other liquids or gases, the effect may be greater.

Last edited: May 5, 2015