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

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

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:
> <nek5000-users at lists.mcs.anl.gov>
> Date:
> 07/27/2017 10:22 PM
>
> To:
> "nek5000-users at lists.mcs.anl.gov" <nek5000-users at lists.mcs.anl.gov>
>
>
>
> 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 <nek5000-users-bounces at lists.mcs.anl.gov> on 
> behalf of nek5000-users at lists.mcs.anl.gov 
> <nek5000-users at lists.mcs.anl.gov>
> *Sent:* Thursday, July 27, 2017 2:11 AM
> *To:* nek5000-users at lists.mcs.anl.gov
> *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
>
> Kaiserstraße 10
> Building 10.23
> 76131 Karlsruhe, Germany
>
> Phone: +49 721 608-43027
> E-mail: steffen.straub at kit.edu
> Web: http://www.istm.kit.edu
>
> KIT – The Research University in the Helmholtz Association
>
> Since 2010, the KIT has been certified as a family-friendly university.

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