Hi Paul, thank you very much for the detailed problem description and the laminar test case. Best Regards, Steffen

Subject: Re: [Nek5000-users] Only neumann and periodic boundary conditions for energy equation From: <[email protected]> Date: 07/27/2017 10:22 PM

To: "[email protected]" <[email protected]>

Hi Steffen,

The attached file shows how to deal with the case

you're interested in.

Please see the README, the .pdf, and the .usr files.

hth,

Paul

------------------------------------------------------------------------ *From:* Nek5000-users <[email protected]> on behalf of [email protected] <[email protected]> *Sent:* Thursday, July 27, 2017 2:11 AM *To:* [email protected] *Subject:* [Nek5000-users] Only neumann and periodic boundary conditions for energy equation 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

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