A priori tests of eddy viscosity models in square duct flow
Davide Modesti
Abstract
Abstract We carry out a priori tests of linear and nonlinear eddy viscosity models using direct numerical simulation (DNS) data of square duct flow up to friction Reynolds number $${\text {Re}}_\tau =1055$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mtext>Re</mml:mtext><mml:mi>τ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1055</mml:mn></mml:mrow></mml:math> . We focus on the ability of eddy viscosity models to reproduce the anisotropic Reynolds stress tensor components $$a_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub></mml:math> responsible for turbulent secondary flows, namely the normal stress $$a_{22}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>22</mml:mn></mml:msub></mml:math> and the secondary shear stress $$a_{23}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>23</mml:mn></mml:msub></mml:math> . A priori tests on constitutive relations for $$a_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub></mml:math> are performed using the tensor polynomial expansion of Pope (J Fluid Mech 72:331–340, 1975), whereby one tensor base corresponds to the linear eddy viscosity hypothesis and five bases return exact representation of $$a_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub></mml:math> . We show that the bases subset has an important effect on the accuracy of the stresses and the best results are obtained when using tensor bases which contain both the strain rate and the rotation rate. Models performance is quantified using the mean correlation coefficient with respect to DNS data $${\widetilde{C}}_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub></mml:math> , which shows that the linear eddy viscosity hypothesis always returns very accurate values of the primary shear stress $$a_{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:math> ( $${\widetilde{C}}_{12}>0.99$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mn>12</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0.99</mml:mn></mml:mrow></mml:math> ), whereas two bases are sufficient to achieve good accuracy of the normal stress and secondary shear stress ( $${\widetilde{C}}_{22}=0.911$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mn>22</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.911</mml:mn></mml:mrow></mml:math> , $${\widetilde{C}}_{23}=0.743$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mn>23</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.743</mml:mn></mml:mrow></mml:math> ). Unfortunately, RANS models rely on additional assumptions and a priori analysis carried out on popular models, including k – $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> and $$v^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> – f , reveals that none of them achieves ideal accuracy. The only model based on Pope’s expansion which approaches ideal performance is the quadratic correction of Spalart (Int J Heat Fluid Flow 21:252–263, 2000), which has similar accuracy to models using four or more tensor bases. Nevertheless, the best results are obtained when using the linear correction to the $$v^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> – f model developed by Pecnik and Iaccarino (AIAA Paper 2008-3852, 2008), although this is not built on the canonical tensor polynomial as the other models.