## quinta-feira, 4 de novembro de 2010

### Conservation of Energy for LES

In order to find the energy dissipation rate, it is essential to understand the way in which kinetic energy is transferred between the filtered velocity field and the residual motions. "The filtered kinetic energy E ̅(x,t) is obtained by filtering the kinetic energy field E(x,t)≡1/2 (U∙U) ̅: E ̅≡1/2 (U∙U) ̅ (1)
It can be further decomposed as:
E ̅≡E_f+k_r (2)
where E_f, the kinetic energy of the filtered velocity field, and k_r, the residual kinetic energy, are given by
E_f≡1/2 U ̅∙U ̅ (3)
k_r≡1/2 (U∙U) ̅-1/2 U ̅∙U ̅=1/2 τ_ii^R (4)
The conservation equation for Ef is:
(D ̅E_f)/(D ̅t)-∂/(∂x_i ) [U ̅_j (2υS ̅_ij-τ_ij^r-p ̅/ρ δ_ij ) ]≡-ε_f-P_r (5)
where ε_f and P_r are defined as
ε_f≡2υS ̅_ij S ̅_ij (6)
P_r≡-τ_ij^r S ̅_ij (7)

where S ̅_ij is the filtered rate of strain and τ_ij^r is the anisotropic residual-stress tensor and defined as:
τ_ij^r≡τ_ij^R-2/3 k_r δ_ij (8)
where δ_ij, the kronecker delta, and τ_ij^R, the residual –stress tensor, are defined as:
δ_ij=1,when i=j and=0,when i≠j (9)
τ_ij^R≡(U_i U_j ) ̅-(U_i ) ̅(U_j ) ̅ (10)
In Equation 5, terms of the left hand side represents the transport and the ones on the right hand side represents the sink terms. The first sink term -ε_f corresponds to the viscous dissipation directly from the filtered velocity, and at high Reynolds number with filter width much larger than the Kolmogorov scale, this term is relatively small. The second sink term 〖-P〗_r, which appears as a source term (〖+P〗_r) in the equation for kr, is the rate of production of residual kinetic energy in the equation of E_f; therefore, it represents the rate of transfer of energy from the filtered motion to the residual motions. Sometimes, it is referred as the sub-grid scale (SGS) dissipation and denoted by εs.
At high Reynolds number the filtered velocity field accounts for nearly all of the kinetic energy:
E ̅ ≈ E (11)
As seen above, the dominant sink for E ̅ is P_r , whereas that in the equation for E for two-equation models is the rate of dissipation of kinetic energy, ε; consequently:
P_r≈ε (12)
Hence,
ε≈P_r≡-τ_ij^r S ̅_ij (13)
Now, both terms S ̅_ij and τ_ij^r can be obtained directly from the transient results file and they need to be averaged with time.
there are two different kinds of energy dissipation rates: solved and SGS.
Solved, which is comparatively much smaller and can be neglected, is represented by the term ε_f, and SGS is represented by〖 P〗_r, which is the dominant term and accounts for nearly all of the energy dissipation.
Sij is the shear strain rate with units s-1.
τ_ij^r≡τ_ij^R-2/3 k_r δ_ij
τ_ij^R≡(U_i U_j ) ̅-(U_i ) ̅(U_j ) ̅
k_r≡1/2 (U∙U) ̅-1/2 U ̅∙U ̅=1/2 τ_ii^R

Note 1: these Equations are written in Linear method. If these difficult to understand then copy and paste in MS word 2007 (not sure about the other ones, you can try it) and convert these equations to Professional.
Note 2: This is based on the understanding of the equations from the book Turbulent Flows by Stephen B Pope. Italicized is direct copy and the rest is adapted form Turbulent Flows by Stephen B Pope.