Thermal boundary layer and hydrodynamic boundary layer

[Urmi Roy]

So I know individually how these form. Unfortunately I haven't found any sources that describe more detailed questions that pop up in my mind.

Could someone help me answer a couple of questions?

1. So if a thermal boundary layer forms in a 'plug flow' model i.e. when there is no momentum b.;ayer formation, the temperature near the surface assumes a parabolic profile.

However, how does the formation of a momentum b.l affect the thermal b.l? Do they form independent of each-other?

2. In a thermal boundary layer, why is it that we define the thickness of the boundary layer as the thickness required to get a (T-Ts)=0.99*T∞ where Ts=surface temperature; T=temp at ∂t and T∞ is free stream temperature. (in momentum bl, boundary layer thickness is when velocity becomes 0.99*U∞)

 

[Chestermiller]

So I know individually how these form. Unfortunately I haven't found any sources that describe more detailed questions that pop up in my mind.

Could someone help me answer a couple of questions?

1. So if a thermal boundary layer forms in a 'plug flow' model i.e. when there is no momentum b.;ayer formation, the temperature near the surface assumes a parabolic profile.

There is no momentum bl formation because the velocity profile is already fully developed before the heated section of wall is encountered. The temperature near the wall does not assume a parabolic profile, unless you approximate it by a parabola, and use the energy integral method. The more exact profile is a complementary error function variation (at least for the case of plug flow).

However, how does the formation of a momentum b.l affect the thermal b.l? Do they form independent of each-other?

If we neglect the effect of temperature on viscosity, the momentum bl forms independently of the temperature bl, and is somewhat thicker than the temperature bl. However, since the velocity profile associated with a developing momentum boundary layer is changing with distance along the wall, this affects the development of the temperature boundary layer (which is forming inside the momentum boundary layer).

2. In a thermal boundary layer, why is it that we define the thickness of the boundary layer as the thickness required to get a (T-Ts)=0.99*T∞ where Ts=surface temperature; T=temp at ∂t and T∞ is free stream temperature. (in momentum bl, boundary layer thickness is when velocity becomes 0.99*U∞)

It's not given by your equation. It's given by (T-Ts)=0.99*(T∞-Ts). Mathematically, the temperature profile describing the thermal boundary layer extends to infinity. However, on a practical basis, the temperature reaches the free stream temperature within only a short distance from the wall. This is approximated by saying that, once the temperature rise is within 1% of (T∞-Ts), you are essentially there. There are other definitions for the boundary layer thickness similar to this one.

 

[Urmi Roy]

Thanks, that was a very enlightening post

Just two sources of confusion here:

There is no momentum bl formation because the velocity profile is already fully developed before the heated section of wall is encountered. The temperature near the wall does not assume a parabolic profile, unless you approximate it by a parabola, and use the energy integral method. The more exact profile is a complementary error function variation (at least for the case of plug flow).

Firstly, my impression of fully developed flow was that there's still a boundary layer, just that it's reached its maximum thickness, since the flow conditions no longer vary beyond a certain length of the tube. However, you mentioned that "There is no momentum bl formation because the velocity profile is already fully developed"

Secondly, in certain problems we did, we approximated the boundary layer to be like the semi-infinite body heating problem (in conduction) when its in its very initial stage. Its when this is true that the b.l takes the shape of the erfc function. That approximations shouldn't be applicable beyond some point (when the b.l is more developed, I guess and the profile of T is parabolic)...so why is it that for plug flow model the b.l is always erfc-like?

Thanks!

 

[Chestermiller]

Thanks, that was a very enlightening post

Just two sources of confusion here:

Firstly, my impression of fully developed flow was that there's still a boundary layer, just that it's reached its maximum thickness, since the flow conditions no longer vary beyond a certain length of the tube. However, you mentioned that "There is no momentum bl formation because the velocity profile is already fully developed"

If you want to consider the parabolic velocity profile for fully developed laminar flow in a circular tube as boundary layer that has penetrated to the center of the tube and is no longer changing, I have no problem with that.

When I said that "There is no momentum bl formation because the velocity profile is already fully developed," what I meant was that the velocity profile is fully established (either for laminar flow or plug flow) before the fluid reaches the heated section of pipe.

Secondly, in certain problems we did, we approximated the boundary layer to be like the semi-infinite body heating problem (in conduction) when its in its very initial stage. Its when this is true that the b.l takes the shape of the erfc function. That approximations shouldn't be applicable beyond some point (when the b.l is more developed, I guess and the profile of T is parabolic)...so why is it that for plug flow model the b.l is always erfc-like?

You are correct in saying that, for plug flow, the thermal boundary layer starts to develop like the erfc solution. However, the shape of the temperature profile, at least for the constant wall temperature case, never approaches a parabolic profile. In plug flow in a circular tube, the erfc solution becomes inaccurate once the erfc boundary layer thickness grows to the point where it approaches the center of the tube. After that, it is not a good approximation, and the behavior approaches the asymptotic temperature profile solution that you get by solving the transient heat conduction equation at long times (using products of exponential time functions times trigonometric spatial functions). This profile is not parabolic.

Chet

 

[Urmi Roy]

Thanks, this makes sense!