Constitutive Relations and Realizability of Single-Point Turbulence Closures

Abstract

In turbulent computational fluid dynamics (CFD), the basic equations are the ensemble averaged Navier-Stokes equations, such as the mean velocity and mean scalar (e.g., temperature) equations, the Reynolds stress and turbulent scalar flux equations, etc. These transport equations are, in general, not closed because of the new unknown correlation terms created by the non-linearity of the Navier-Stokes equation in the statistical averaging process. These new unknown terms are onepoint (in time and space) turbulent correlations, for example, the Reynolds stress EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WG1bWaaSbaaSqaaiaadMgaaeqaaOGaamyDamaaBaaaleaacaWGQbaa % beaaaaaaaa!3A36!EquationSource$$ \[\overline {{u_i}{u_j}} \]$$ and the scalar flux EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca %WG1bWaaSbaaSqaaiaadMgaaeqaaOGaeqiUdehaaaaa!39D7!EquationSource$$ \[\overline {theta\{u_i} } \]$$ in the mean velocity U _i and mean scalar Θ equations, or the pressure-strain correlation EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WG1bWaaSbaaSqaaiaadMgaaeqaaOGaamyDamaaBaaaleaacaWGQbaa % beaaaaaaaa!3A36!EquationSource$$ \[\overline {{p}({u_i,j} + {u_j,i})} \]$$ and the pressure-scalar gradient correlation EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca %WG1bWaaSbaaSqaaiaadMgaaeqaaOGaeqiUdehaaaaa!39D7!EquationSource$$ \[\overline {p\{theta_,i} } \]$$ in the Reynolds stress and scalar flux transport equations. These terms and other unknown correlation terms (e.g., the triple correlations EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WG1bWaaSbaaSqaaiaadMgaaeqaaOGaamyDamaaBaaaleaacaWGQbaa % beaaaaaaaa!3A36!EquationSource$$ \[\overline {{u_i}{u_j}{u_k}} \]$$, EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WG1bWaaSbaaSqaaiaadMgaaeqaaOGaamyDamaaBaaaleaacaWGQbaa % beaaaaaaaa!3A36!EquationSource$$ \[\overline {{u_i}{u_j}{theta}} \]$$ and EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WG1bWaaSbaaSqaaiaadMgaaeqaaOGaamyDamaaBaaaleaacaWGQbaa % beaaaaaaaa!3A36!EquationSource$$ \[\overline {{u_i}{theta^2}} \]$$) must be modeled in order to close the corresponding CFD equations and make them ready to be used for studying various turbulent flows. The task of one-point closures is to provide models or “constitutive” relationships for these new unknown one-point turbulent correlation terms. In this chapter we shall follow the ideas and approach of Lumley (1970) in deriving the possible general constitutive relationships for various one-point turbulent correlations. Then, we shall discuss the realizability concept introduced by Schumann (1977) Lumley (1978) and Reynolds (1987) and show its application in various levels of one-point turbulence closures for both velocity and scalar fields.