Forced convection around a cylinder

We consider the forced convection of an heat source at the entrance of a channel with a cylinder inside.

1. Running the case

The command line to run this case is

mpirun -np 4 feelpp_toolbox_heatfluid --case "github:{repo:toolbox,path:examples/modules/heatfluid/examples/TurekHron}"
mpirun
Case option
--case "github:{repo:toolbox,path:examples/modules/heatfluid/examples/TurekHron}"
The report of the execution of the command above is available here.

2. Data files

The case data files are available in Github here

3. Geometry

A channel with a cylinder inside

We consider a 2D model representative of a laminar incompressible flow around an obstacle. The flow domain, named ΩfΩf, is contained into the rectangle [0,Long]×[0,Haut][0,Long]×[0,Haut]. It is characterised, in particular, by its dynamic viscosity μfμf and by its density ρfρf.

Geometry of the Turek & Hron HeatFluid Benchmark

TurekHron Geometry

In order to describe the flow, the incompressible Navier-Stokes model is chosen for this case, define by the conservation of momentum equation and the conservation of mass equation. At them, we add the material constitutive equation, that help us to define σf

The goal of this benchmark is to couple the Naviers-Stockes equations and the heat equations We remind that the Naviers-Stokes equation are

ρ(ut+uu)νΔu+p=f  in Ωu=0  in Ω

And the Heat equations is

ρCp(Tt+uT)(kT)=Q, in Ω

The toolbox is HeatFluid

4. Input parameters

The following table displays the various fixed and variables parameters of this test-case.

Table 1. Fixed and Variable Input Parameters

Name

Description

Units

u

fluid velocity

m/s

ρ

density of the fluid

kg/m3

ν

dynamic viscosity

kg/(m×s)

p

pression

Pa

f

source term

kg/(m3×s)

Cp

thermal capacity

J/(kgK)

T

Temperature

K

Q

heat source

W.m3

k

Thermal conductivity

Wm1K1

4.1. initial condition

For the fluid:

We use a parabolic velocity profile, in order to describe the flow inlet by Γin, which can be express by

vcst=1.5ˉU40.1681y(0.41y)

where ˉU is the mean inflow velocity.

However, we want to impose a progressive increase of this velocity profile. That’s why we define

vin={vcst1cos(π2t)2 if t<2vcst otherwise 

With t the time.

For the temperature:

We give as source this temperature

Tin=300(y>0.15)(y<0.25)+(293.15(y<(0.151e9)))+(293.15(y>(0.251e9)))

4.2. Materials

    "Materials":
    {
        "Fluid":{
            "rho":"1.0e3",
            "mu":"1.0",
            "k":"2.9e-5",
            "Cp":"4185"
        }
    },
json

4.3. Boundary conditions

For the fluid:

We set

  • On Γin, an inflow Dirichlet condition : uf=(vin,0)

  • On Γwall and Γobst, a homogeneous Dirichlet condition : uf=0

  • On Γout, a Neumann condition : σfnf=0

For the heat:

  • On Γin, an inflow Dirichlet condition : Tf=Tin

4.4. Fields

We are intersting in the visualisation of the three fields : the velocity, the pressure and the temperature of the fluid

            "Exports":
            {
                "fields":["fluid.velocity","fluid.pressure","heat.temperature","fluid.pid"]
            },

4.5. Measures

the pressure is measured on two points to see the behavior of the pressure as a function of time

            "Measures":
            {
                "Forces":"wall2",
                "Points":
                {
                    "pointA":
                    {
                        "coord":"{0.6,0.2,0}",
                        "fields":"pressure"
                    },
                    "pointB":
                    {
                        "coord":"{0.15,0.2,0}",
                        "fields":"pressure"
                    }
                }
            },
json

5. Numerical Experiments

We run this model, using the command labeled at the top, we have the following results.

Temperature:
Velocity
Pressure: