# Vertical Heated Pipe

Authors: J. You et al.

Type: Numerical

Status:

Description

## Description

• Low-Reynolds RANS models are compared to DNS and LES data for profiles, and a number of experiments showing Nusselt versus heat/buoyancy loading.
• Nusselt Number versus Buoyancy:
• What it shows: In the heated & upward flow case the near wall layer is accelerated by buoyancy. This shifts the high velocity gradient region closer to the wall and as a result turbulence production is restrained by wall proximity, and eventually the flow relaminarises. It is a very good test of whether models correctly account for interaction between the actual turbulence length-scale (size of large eddies), and the non-local influence of a solid wall. The V2F , Lien & Leschziner, and Launder & Sharma models perform well. The k-omega, and SST models miss the relaminarisation. There is some similarity with accelerating boundary layers (see TaniaKlein 1st year report in this Twiki), as there is no direct effect of buoyancy on the turbulence.

* Key results: Some models (e.g. k-omega SST) miss the relaminarisation * Velocity: = = Temperature: = = Kinetic energy:

## Flow Parameters

Re = 2650 (based on radius)

Pr=0.71

• %$Re = 2650$% (based on radius) %$\Rightarrow Re_\tau=180$%
• GRashof number %$Gr=\frac{g \beta q_w R^4}{\lambda \nu^2}$%
• Constant properties with the exception of the density (isobaric):
• %$\rho = \frac {\rho_0} {1+\beta(T-T_0)}$% where %$\rho_0 = 1.205$%, %$\beta =0 .00343$% and %$T_0 = 293$%
• %$\mu = 18.1 \cdot 10^{-6}$%
• %$\nu = \mu/\rho_0$%
• %$c_p = 1006$%
• %$Pr = 0.71$%
• Five different heat transfer regimes are considered:
• %$\frac {Gr} {Re^2} = 0 \Rightarrow$% "force convection"
• %$\frac {Gr} {Re^2} = 0.063 \Rightarrow$% "force/mixed convection"
• %$\frac {Gr} {Re^2} = 0.087 \Rightarrow$% "re-laminarization"
• %$\frac {Gr} {Re^2} = 0.241 to 0.400 \Rightarrow$% "recovery"

### Geometry and Mesh generation

• The flow is fully developed. Periodicity in the stream-wise direction can be used
• The flow pattern is axial-symmetric. For RANS it is possible to use just one cell in the ortho-radial direction
• The RANS mesh for RANS (See CONVERT in results section)
• n cells: 120
• distribution: geometric progression with expansion factor equal to 0.99, $y^+ \leq 0.3$

### Boundary conditions

• Streamwise direction (z): periodicity, with a momentum source term to drive the flow

### Source terms

In the stream-wise direction a source term corresponding to the bulk pressure gradient is imposed to drive the flow. It can be definded in two differents ways.

1. Pressure Drop

%BEGINLATEX{label="bal_mom"}%

\begin{equation*}p S_{inlet}-(p+dp)S_{outlet}=\tau_wS_{lat} \end{equation*}

%ENDLATEX%

%BEGINLATEX{label="pres_drop"}% \begin{equation*} S_{inlet}=S_{outlet}=S \Rightarrow dp=-\tau_w \frac {S_{lat}} {S} = -\tau_w \frac {P dz} {S} \end{equation*} %ENDLATEX% %BEGINLATEX{label="beta"}% \begin{equation*} \beta= \frac {dp} {dz} = -(\frac {2\tau_w} {R})_{\mbox{for\,pipe}} \end{equation*} %ENDLATEX% the term %$\beta$% has to be added into the '''explicit''' part of the source term into the z direction, with a positive sign.

1. Mass flow rate
%BEGINLATEX{label="mass_rate"}% \begin{equation*} \dot{m}_{id} = \frac {Re \mu S_{inlet}} {R} \end{equation*} %ENDLATEX% At every iteration the mass flow rate at the outlet (%$\dot{m}_{calc}$%) has to be computed. The source term has the following formulation: %BEGINLATEX{label="err_mass"}% \begin{equation*} err=\dot{m}_{id}-\dot{m}_{calc} \end{equation*} %ENDLATEX% It has to be insert into the '''implicit''' part of the source term.

Also for the energy equation a source term is required. If we consider the element of figure-1 and we apply the first principle of thermodynamic from the point of view of an external observer moving with the fluid and we also consider constant density we can write: %BEGINLATEX{label="1_princ_termo"}% \begin{equation*} dE=dQ \end{equation*} %ENDLATEX% and from this equation it is possible to compute the variation of the bulk temperature between inlet and outlet as: %BEGINLATEX{label="Delta_T"}% \begin{equation*} T_{b_{outlet}}- T_{b_{inlet}}=\frac {\dot{q}S_{wall}} {\dot{m}c_p} \end{equation*} %ENDLATEX% Then the source term for the transport equation of the temperature is the following: %BEGINLATEX{label="source_term"}% \begin{equation*} s_T=-\rho W \sigma \end{equation*} %ENDLATEX% where %$\sigma=\frac {dT} {dz} = \frac {\dot{q}S_{wall}} {\dot{m}c_pdz}$%

## Reference Publications

1. You et al 2003; "Direct numerical simulation of heated vertical air flows in fully developed turbulent mixed convection"; International Journal of Heat and Mass Transfer, Volume 46, Issue 9, April 2003, Pages 1613-1627. Download
2. Keshmiri A., Addad Y., Cotton M.A., Laurence D. and Billard F., 2008; "Refined Eddy Viscosity Schemes and LES. for Ascending Mixed Convection Flows" To appear in the Proc. of Computational Heat Transfer (CHT08) Symp., Marakkech, Morroco, 11-16 May 2008 Download
3. Keshmiri A., Cotton M.A., Addad Y., Rolfo S. and Billard F., 2008,; "RANS and LES Investigations of Vertical Flows in Passages of Gas-Cooled Nuclear Reactors" To appear in the Proc. of ASME 16th Int. Conf. on Nuclear Engineering (ICONE16), Orlando, Florida, 11-15 May 2008, Vol. ICONE16-48372 Download
4. Keshmiri A. and Cotton M.A., 2008,; "Turbulent Mixed Convection Flows Computed Using Low-Reynolds-Number and Strain Parameter Eddy Viscosity Schemes"; Proc. of 7th Int. ERCOFTAC Symp. on Engineering Turbulence Modelling and Measurements (ETMM7), Vol. 1, pp.274-279, Limassol, Cyprus, 4th-6th June 2008.Download
5. W.S. Kim, S. He and J.D. Jackson, 2008; "Assessment by comparison with DNS data of turbulence models used in simulations of mixed convection"; International Journal of Heat and Mass Transfer, Volume 51, Issues 5-6, , March 2008, Pages 1293-1312. download doi ijheatmasstransfer.2007.12.002

## Results

Simulation results available for this case:
Code Version Author Restrictions
CONVERT [[CfdTm.TestCase005Res001][]] Amir Keshmiri Main.None
CONVERT [[CfdTm.TestCase005Res002][]] Amir Keshmiri Main.None
CONVERT [[CfdTm.TestCase005Res003][]] Amir Keshmiri Main.None
reference DNS [[CfdTm.TestCase005Res004][]] You et al [2003] Main.None
Code_Saturne [[CfdTm.TestCase005Res006][]] Flavien Billard Main.None
STAR-CD, k-eps [[CfdTm.TestCase005Res007][]] Stefano Rolfo Main.None
STAR-CD, SST [[CfdTm.TestCase005Res008][]] Stefano Rolfo Main.None
Number of topics: 8

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Topic attachments
I Attachment Action Size Date Who Comment
ppt Heated-PipeShort_KNOO_Addad_Laurence.ppt manage 2396.0 K 2008-11-23 - 13:57 DominiqueLaurence More graphs in ppt file
JPG ICONE16-Nu.JPG manage 137.4 K 2009-02-12 - 17:22 AmirKeshmiri
png slide0012_image032.png manage 50.3 K 2008-11-23 - 12:50 DominiqueLaurence Velocity
png slide0012_image033.png manage 51.5 K 2008-11-23 - 12:51 DominiqueLaurence Temperature
png slide0012_image034.png manage 65.8 K 2008-11-23 - 12:53 DominiqueLaurence Kinetic energy
Topic revision: r49 - 2011-02-10 - 09:08:09 - StefanoRolfo
CfdTm Web
22 Sep 2019

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