[Nek5000-users] Only neumann and periodic boundary conditions for energy equation

[nek5000-users at lists.mcs.anl.gov]

Hi Neks,

I want to simulate "ideal" isoflux boundary conditions in a pipe.
As the temperature (T) increases in streamwise direction (in my case 
z-direction), I define
theta(r,phi,z,t) = (<T_w>(z) - T(r,phi,z,t))/T_r
with T_r = q_w/(rho c_p U_b) and <T_w>(z) denoting the time average of 
the wall temperature T_w.

With that I can recast my energy equation to solve for the temperature 
difference theta instead of the temperature T which allows for periodic 
boundary conditions as theta does not change in streamwise direction. 
This introduces an additional source term 4 u_z.

I would like to set a constant heat flux boundary condition at the wall 
(see e.g. Piller: Direct numerical simulation of turbulent forced 
convection in a pipe. 2005), i.e. a Neumann boundary condition (ideal 
isoflux), and compare the results to those obtained with the same PDE 
but a Dirichlet boundary condition theta_w=0 (mixed-type). This setup of 
applying only Neumann boundary conditions is "ill-posed". I believe 
because there is no unique solution to this setup, right?
As Piller points out, one can introduce an additional constraint and 
enforce the volume averaged temperature to be constant to overcome this 
issue. Piller did not face this problem as he was using a finite volume 
method.

I can calculate the volume integral over temperature like this, correct?
       nt = nx1*ny1*nz1*nelt
       t_vol = glsc2(t, bm1, nt)

And then I would adjust my source term in each step to keep t_vol=constant.
However, I do not know this constant in advance. If I set it to an 
arbitrary value, e.g. zero, this leads to negative theta at the wall, 
which contradicts my definition of theta.


I know this is not a specific Nek5000 Problem but maybe someone has 
experienced similar issues and found a solution that works in Nek5000?

Best Regards,
Steffen Straub

-- 
Karlsruhe Institute of Technology (KIT)
Institute of Fluid Mechanics

M.Sc. Steffen Straub
Doctoral Researcher

Kaiserstraße 10
Building 10.23
76131 Karlsruhe, Germany

Phone: +49 721 608-43027
E-mail: steffen.straub at kit.edu
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