UFR 3-06 Test Case

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Natural and mixed convection boundary layers on vertical heated walls (A)

Underlying Flow Regime 3-06               © copyright ERCOFTAC 2004

Test Case

Brief description of the study test case

A variety of basic studies has been conducted examining turbulence structure in simple boundary layers with constant properties and uniform wall temperatures. A strong heating atthe wall induced by a nearly constant high heat flux boundary condition involves significant transport property variation and buoyancy effects and makes necessary to further assess the predictive capabilities of turbulence models in that situation. Experiments for air flowing upwards in small heated vertical tubes with temperature dependent transport properties were made by Humble, Lowdermilk and Desmon (4), Jackson (5), McEligot, Magee and Leppert (6), Perkins and Worsoe-Schmidt (7) and others. These studies were lacking of internal profile measurements for temperature and velocities but they provided some average data to make preliminary tests of turbulence models accounting for temperature-dependent transport properties and the influence of intense heating on turbulent flows in tubes. 

In (3) Shehata and McEligot have provided new data on mean velocity profiles along with mean temperature profiles, wall temperature and axial pressure distributions in a carefully controlled experiment. These data have been compared in (1) to the computational results of a number of two equations turbulence models for turbulence kinetic energy and its dissipation rate <ε or the related quantity ω. The definition of this UFR makes an extensive use and summarizes these two recent publications.

Non dimensional Parameters

The Reynolds number Re = G*D/μ and the non dimensional heating rate q+in = q”w/G*Cpin*Tinevolve naturally from non dimensionalizing the governing equations and boundary conditions in pipe flow with an imposed wall heat flux distribution (8).G is the mean mass flux (mass flow-rate divided by the cross sectional area) D is the tube diameter, μ the absolute viscosity q” is the prescribed heat flux at the wall, , Cp the specific heat at constant temperature, T the absolute temperature. 

The “in” index means that the value is related to the inlet. The “w” index means that the value is related to the wall. The experimental data reported in (3) are available for the following set of parameters: Low Reynolds numbers (4260 and 6080), moderate, high and intense heating rates (q+in = 0.0018, 0.0035, 0.0045) yielding flows ranging from turbulent to laminarized 

2.1 Computational Studies of Mixed Convection in circular tubes with intense heating

The various two equations models used in the study reported in (1) are of the Low Reynolds number (k, ε) type, they differ in respect of the coefficients of the diffusive terms and also in respect of additional source terms which are introduced in some of them to account for pressure diffusion of k and production of ε.

These models are detailed in the references cited hereunder:JL: Low Reynolds number (k, ε) turbulence model proposed by Jones and Launder(10)LS: Low Reynolds number (k, ε) turbulence model proposed by Launder and Sharma (11)CH: Low Reynolds number (k, v) turbulence model proposed by Chien (12)LB: Low Reynolds number (k, ε) turbulence model proposed by Lam and Bremhorst (13)MRS: Low Reynolds number (k, ε) turbulence model proposed by Michelassi et al. (14)SH: Low Reynolds number (k, ε) turbulence model proposed by Shi and Hsu (15)TAS: Low Reynolds number (k, τ) turbulence model proposed by Thangan et al (16) 

2.2 Choice of UFR

Three experiments described Shin-ichi Satake, Tomoaki Kunugi, Mohsen Shehata and D.M.McEligot in (3) are chosen as a test case. They are called runs 445, 635 and 618. Run 445 is for Re= 4260 , q+in = 0.0045Run 635 is for Re= 6080 , q+in = 0.0035 Run 618 is for Re= 6080 , q+in = 0.0018 

These runs are used because they cover a wide range of heating rates and the Reynoldsnumber is representative of the Reynolds number used in real power generation units. Theb flow conditions at the entrance of the heated section are fully established as it comes after a long enough adiabatic section. The wall heat flux distribution is given with a claimed accuracy of 3%. 

2.3 Brief description of the test case

The test case consists of upward flow of air in a heated circular tube. The inside diameter of the tube is 27 mm. The unheated length before the heated test section is 50 diameters long(1.35 m.) and the heated test section has a length of 32 diameters (0.86 m.). 

The parameters that define the flow regime are the inlet Reynolds number, the nondimensional heating rate (based on wall heat flux), the inlet pressure and gas temperature distribution and the wall temperature. The principal derived quantities, by which the success, or failure, of CFD calculations is judged, are the predicted wall temperatures compared to measurements and the predicted axial development of mean streamwise velocity and mean temperature compared to the measurements at three to five elevations. No measurement of any turbulent quantity is available.

Test Case Experiments

The basic test section is the same for all cases. A resistively heated, seamless, smooth, extruded Inconel 600 tube of 27.4 mm inside diameter was employed as the test section. The short heated length was picked to permit high heating rates with this material while possibly approaching quasi developed conditions. Outside wall temperatures were determined with premium grade Chromel Alumel thermocouples distributed along the tube. The axial variation of the static pressure was obtained with pressure taps electrostatically drilled through the wall. A single hot wire sensor has been selected to measure the streamwise velocity and temperature in order to minimize flow disturbances and to permit measurements close to the wall. It was employed as a hot wire for velocity measurements and as a resistance thermometer for pointwise temperatures. 

The heat losses were calibrated so the consequent uncertainties in the wall heat flux were below 3%. The uncertainty in velocity was calculated to be in the range of 8-10% of the pointwise value, with the larger percent uncertainties occurring near the wall. The uncertainty in temperature was typically 1-2% of the pointwise absolute temperature. These estimates are believed to be conservative since comparisons of the integrated and measured total mass flow rates for each profile showed better agreement, of the order of 3% or less, except near the tube exit.

CFD Methods

The computer program used in the study reported here is a version of the CONVERT code(9). The governing equations are those of steady, single phase axisymetric flow in a vertical tube with no swirl and no flow reversal. The governing equations are cast in the “internal boundary layer” approximation which reduces significantly the number of terms appearing in the conservation equations and permits numerical solution by a marching technique. 

The mean flow equations for conservation of mass, momentum and thermal energy arewritten in the “thin shear layer” form which results from considering flow where there is a clear principal flow direction and the main variation of velocity occurs in the direction normal to this direction. In the duct geometry considered, a circular tube, the principal flow direction coincides with the axis OZ and velocity gradients in the radial direction are dominant. 

For the two equations turbulence models, the equations used describe the transport of turbulence kinetic energy and its rate of dissipation (Table1).Transport of turbulent kinetic energy and its dissipation rate: The integral continuity equation (conservation of mass flow rate) is the final equation relating the dependent variables: the two velocity components W, V, the enthalpy h, the turbulent kinetic ienergy k, the dissipation rate e and the pressure p.

Table 1 (Ref.1): the “internalboundary layer“ equations for a vertical tube.

The condition of no slip is applied at the tube wall. The thermal condition at the wall is a specified axial distribution of wall heat flux (near uniform value is obtained in a few diameters). 

The inlet conditions imposed are specified fully developed profiles of the streamwise velocity, turbulence kinetic energy and dissipation and a uniform temperature profile. Preliminary calculations for an unheated tube were performed to generate these profiles.

4.1 Turbulence models examined

The various two equations low Reynolds turbulence models which have been examined in(xx) differ in respect of the functions f_m, f2 and f3 which are used and the additional terms Pand E introduced to account for pressure diffusion of k and production of e respectively, in the near wall region. 

For all models, a value of 0.9 is chosen for the turbulent Prandtl number st. The constant Cmis equal to 0.09 for all models except TAS for which the value is 0.096. C1 is equal to 1.44 inLS, LB, MRS and TAS models, 1.55 in JL, 1.35 in Chien and 1.5 in SH model. C2 is equal to1.92 in LS, LB and MRS, 2.0 in JL and SH, 1.8 in Chien, 1.83 in TAS model. sk is equal o1.0 in LS, JL, Chien, LB, 1.3 in SH and MRS, 1.36 in TAS. s_e is equal to 1.3 for all models except TAS where the value is 1.36. 

The functions f _m, f2 and f3 and the terms P and E are listed for each model in Table 2. No wall functions are used, the (k, ε) equations are solved in the entire domain,k=0 is the boundary condition at the wall, the wall boundary conditions for e were evaluated as shown in Fig.2., following the recommendations of the models' authors.

4.2 Computational grid and grid independence

Non uniform distributions are employed for the node spacing in both radial and axial directions. A fine grid is needed in the near wall region to resolve steep gradients in the mean and turbule nce fields. 101 nodes in the radial direction are assigned with the first at y+ = 0.5 and the 51th at y+ = 30. In the streamwise direction step sizes are based on Dz equals approximately two times the viscous sublayer thickness. 

Numerical accuracy of the calculation procedure has been examined by conducting sensitivity tests and making comparisons with accepted results. The sensitivity tests involved varying radial and axial spacing of nodes, convergence criteria, relaxation factors and number of internal iterations. The changes observed with the most stringent conditions never changed parameters such as the Nusselt number, friction factor and centerline velocity more than 0.1%. 

Table 2 (from ref.1): Definitions of functions and boundary conditions for (k, ε) models © copyright ERCOFTAC 2004

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Contributors: André Latrobe - CEA / DRN / Department de Thermohydraulique

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Description

Test Case Studies

Evaluation

Best Practice Advice

References