Conservational PDF equations of turbulence

Original publisher: Cleveland, Ohio : National Aeronautics and Space Administration, Glenn Research Center, [2010] OCLC Number: (OCoLC)698260211 Excerpt: ... U Φ�ψ V f ( V, ψ; x, t ) dV dψ i n n i U, Φ��∫ ∫ ( 34 )�ψ f ( ψ; x, t ) U ψ dψ n Φ i ∫ or U Φ�V ψ f ( V, ψ; x, t ) dψ dV i n i n U, Φ��∫ ∫ ( 35 )�V f ( V; x, t ) Φ V dV i U n ∫ Equations ( 34 ) and ( 35 ) illustrate that, when taking joint mean, the joint PDF must be used to avoid the U ψ appearance of the conditional mean. The conditional mean is also carried by the current i traditional species PDF equation. Finally, let us examine the mean and the conditional mean of a function of the variables Φ, say i S ( Φ Φ Φ ) S ( Φ Φ Φ )��, on the condition of. Let us first view as a separate random Φ�ψ 1 2 n 1 2 n variable with the sample space variable s which is independent of ψ, then, following Equation ( 27 ), we write the mean as S ( ΦΦ�Φ )�f ( ψ; x, t ) S ( ΦΦ�Φ ) ψ dψ ( 36 ) 1 2 n Φ 1 2 n ∫ The conditional mean in ( 36 ) is S ( Φ Φ Φ ) ψ�S ( ψ ψ ψ ) f ( s ψ; x, t ) ds��n n 1 2 1 2 S Φ ∫�ψ�S ( ψ ψ ) f ( s ψ; x, t ) ds ( 37 ) 1 2 n Φ S ∫�ψ�S ( ψ ψ ) 1 2 n Then Equation ( 36 ) will end up an expression that is just the definition of its mean: S ( ΦΦ�Φ�S ( ψψ�ψ ) f ( ψ; x, t ) dψ ( 38 ) 1 2 n 1 2 n Φ ∫ This term is related to the chemical reaction rate in the traditional species PDF equation, which is in a S ( Φ Φ Φ )�closed form. Also note that the reason for the conditional mean of in Equation ( 36 ) 1 2 n S ( Φ Φ Φ )�becoming free from the constraint is that is a known function of Φ without involving i 1 2 n spatial differentiation. 2.4.4 Summary 2.4.4.1 Formulations With the Conditional Mean 2 2 ∇ U�f ( V; x, t ) ∇ U V dV ( 39 ) i U i ∫ ∇ P�f ( V; x, t ) ∇ P V dV ( 40 ) U ∫ 2 2 ∇ Φ�f ( ψ; x, t ) ∇ Φ ψ dψ ( 41 ) i Φ i ∫ NASA / TM - 2010-216368 9