 # Is restricted to wavenumbers within the sphere of ra

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is restricted to wavenumbers within the sphere of ra-diusκc(which is within the cube of side 2κc). Equivalently, the fractionof the represented coeﬃcientsˆu(κ, t) that are non-zero is approximately43πκ3c/(2κc)3=π/6.The Navier–Stokes equations in wavenumber space (Eq. (6.145)on page214) can be writtenddt+νκ2ˆuj(κ, t) =iκ Pjk(κ)κ,κδκ,κ+κˆuk(κ, tu(κ, t),(13.195)wherePjkis the projection tensor (Eq. (6.133)). The Kronecker delta is unityfor triadic interactions, i.e.,κ=κ+κ,(13.196)and is zero otherwise. Equation(13.195)is a coupled set of ordinary differ-ential equations, one equation for each of the infinite number of modes.
60613 Large-eddy simulationThe filtered equation obtained fromEq. (13.195)is, for the non-zerocoeﬃcients (κ < κc),ddt+νκ2ˆuj(κ, t) =iκ Pjk(κ)κ,κδκ,κ+κH(κcκuk(κ, tu(κ, t).(13.197)This is a finite set of (approximatelyπN3/6) ordinary differential equations,but it is unclosed. The closure problem arises because the nonlinear term (onthe right-hand side ofEq. (13.197)) includes unknown Fourier coeﬃcients,namely ˆu(κ) and ˆu(κ) forκ≡ |κ| ≥κcandκ≡ |κ| ≥κc.EXERCISE13.34Consider LES of high-Reynolds-number homogeneous isotropic tur-bulence using the sharp spectral filter and a pseudo-spectral methodwith largest resolved wavenumberκmax=κc. As in DNS, adequateresolution of the large scales requires the sizeLof the domain to beeight integral lengthscales, i.e.,L= 8L11(seeEq. (9.5)). So that nomore than 20% of the energy is in the residual motions, the cutoffwavenumber is taken to beκcL11= 15 (seeTable 6.2on page 240).Show that these requirements determineκmaxκ0=15π/419,(13.198)as the ratio of the resolved wavenumbers, corresponding to a 383simulation.13.5.2 Triad interactionsThere is, of course, a direct connection between the nonlinear term inEq. (13.197), the velocity product in physical spaceuku, and various stressesformed from it (e.g.,uku). FromEq. (13.191)we obtainuk(x, t)u(x, t) =κκei(κ+κ)·xˆuk(κ, tu(κ, t)=κ,κδκ,κ+κeiκ·xˆuk(κ, tu(κ, t).(13.199)Figure 13.10is a sketch of qualitatively different triad interactions. Theseare referred to as types (a)–(d), and are defined inTable 13.4. In terms ofthe modes represented in the LES (i.e.,|κ|< κc), only interactions of type(a) can be represented exactly. Consider the summation inEq. (13.199)for
13.5 LES in wavenumber space607κ'κκκ''c(a)κ'κκ''(b)κ'κκ''κ'κκ''(c)κ'κκ''(d)Fig. 13.10. Sketches of the various types of triad interactions defined inTable 13.4:(a) resolved, (b) Leonard, (c) cross, (d) SGS.ukurestricted to these wavenumbers. The restrictionsκ < κcandκ< κcamount to replacingukbyukandubyu, and the restrictionκ < κcamountsto filtering the result. Thus, the sum of the interactions of type (a) yieldsuku=κ,κδκ,κ+κH(κcκ)H(κcκ)H(κcκuk(κu(κ)=κ,κδκ,κ+κH(κcκ)ˆuk(κ)ˆu(κ).(13.200)
60813 Large-eddy simulationTable 13.4.Definitions of the types of triad interactions sketched inFig. 13.10,and their contributions touku(Eq. (13.199)) (all possible interactions withκ < κcare included, but there are additional interactions withκκc)TypeDefining wavenumberContributiondesignationrangestoukuType (a),κ < κ

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