Defining dimensionless tempearture for Periodic flow systems

Question

Given a flow inside a square duct with constant temperature at the walls $(T_{w1} = T_{w2} = T_w)$ the physical property in terms of temperature that repeats itself in a periodic fashion is the $\textbf{Dimensionless Temperature} ~ \mathbf{\theta}$ defined as shown in the picture below :-

I am interested in defining $\theta$ that is valid for inhomogenous Dirichlet boundary conditions i.e. when $T_{w1} \neq T_{w2}$ which shall show a periodic behavior in fully developed flow.

P.S. In an actual 3D flow through square duct there would be four faces(or walls) and all can be at different fixed temperature. For brevity purpose I have shown a case here which shows two walls in a 2D system (can be considered to be a flow between parallel plates with adibatic/free slip boundary condition for temperature/flow respectively).

EDIT

So I am looking to extend the implementation of streamwise periodic boundary condition for non-equal wall temperatures. In case of equal wall temperatures the $\theta$ can be defined as depicted in the picture which then can be transcended to the periodic boundary condition requirement for Dirichlet conditions to be

$\theta_{0,y,z}$ = $\theta_{L,y,z}$

where $L$ in the fully developed periodic length.

Substituting for $\theta$ and doing simplification one gets

$T_{0,y,z} = c_1 T_{L,y,z} + c_2$

where

$c_1 = \frac{T_b(0) - T_{wall}}{T_b(L) - T_{wall}} $ and $c_2 = T_{wall} (1 - c_1)$

Now for a constant and equal wall temperature using this formulation gives the right nusselt number values as well as temperature values (fixing $T_b(0)$ to be some reference tempearture).

In the case of two constant but, different wall temperatures we have two possibilities

$T_{wall} = T_{wall1}$ or $T_{wall} = T_{wall2}$ or $T_{wall} = avg(T_{wall1}, T_{wall2}) $

Choosing one among the first two makes the temperature bounded by either of the wall temperatures but not both. I observe then a few cells near the wall2 or wall2 where the temperatures go below the cold wall temperature or hotter than the hot wall temperature (depending on which wall is taken as T_{wall} in the expression the problem occurs at the opposite wall). Choosing the third definition, if we see the expression for $c_1$ it is really possible to have a bulk temperature at the outlet to be that of the average wall temperatures which is then a condition of singularity.

So the question remains how should we define $\theta$ so as to have a consistent representation of periodicity and also having a bounded solution as well.

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References

Streamwise periodic boundary conditions for Constant $T_{wall}$

Asymmteric Graez Nusslet problem

Answer 1

You could define $T_w$ as $$T_w=\frac{1}{mes(\partial\Omega)} \int_{\partial \Omega}T_W\,dS$$ Where $T_W$ is the wall temperature and $T_w$ is the average wall temperature.

For a 2D problem with constant wall temperatures, this integral reduces to $$T_w=\frac{1}{2}(T_{W1}+T_{W2})$$ This is so, because in this context $\partial\Omega$ is the upper and lower boundary line of the channel with total length ($mes(\partial\Omega)$) equal to $2L$ and if its temperature is constant the integral over the whole boundary is equal to: $T_{W1}L+T_{W2}L$. Divide them and you will obtain the derired result. For the 3D case it will be straightforward. Now the measure ($mes(\partial\Omega)$) will be the area of a plane, and you will be integrating the wall temperature over this plane...

Or you can simply put $$\theta=T/T_b$$

Edit

What if I say you that the nondimensionalisation with a difference is the same as with a single temperature?

In your definition of $\theta$ change $T\to T+c$, with $c$ arbitrary. You see that this change also results in $T_b\to T_b+c$. If you put $c=T_w$ you will have: $$\theta=T/T_b$$