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Kármán–Howarth equation

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Mathematical equation

In isotropic turbulence the Kármán–Howarth equation (after Theodore von Kármán and Leslie Howarth 1938), which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation.

Mathematical description

Consider a two-point velocity correlation tensor for homogeneous turbulence

R i j ( r , t ) = u i ( x , t ) u j ( x + r , t ) ¯ . {\displaystyle R_{ij}(\mathbf {r} ,t)={\overline {u_{i}(\mathbf {x} ,t)u_{j}(\mathbf {x} +\mathbf {r} ,t)}}.}

For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940,

R i j ( r , t ) = u ′ 2 { [ f ( r , t ) − g ( r , t ) ] r i r j r 2 + g ( r , t ) δ i j } , f ( r , t ) = R 11 u ′ 2 , g ( r , t ) = R 22 u ′ 2 {\displaystyle R_{ij}(\mathbf {r} ,t)=u'^{2}\left\{{\frac {r_{i}r_{j}}{r^{2}}}+g(r,t)\delta _{ij}\right\},\quad f(r,t)={\frac {R_{11}}{u'^{2}}},\quad g(r,t)={\frac {R_{22}}{u'^{2}}}}

where u ′ {\displaystyle u'} is the root mean square turbulent velocity and u 1 ,   u 2 ,   u 3 {\displaystyle u_{1},\ u_{2},\ u_{3}} are turbulent velocity in all three directions. Here, f ( r ) {\displaystyle f(r)} is the longitudinal correlation and g ( r ) {\displaystyle g(r)} is the lateral correlation of velocity at two different points. From continuity equation, we have

∂ R i j ∂ r j = 0 ⇒ g ( r , t ) = f ( r , t ) + r 2 ∂ ∂ r f ( r , t ) {\displaystyle {\frac {\partial R_{ij}}{\partial r_{j}}}=0\quad \Rightarrow \quad g(r,t)=f(r,t)+{\frac {r}{2}}{\frac {\partial }{\partial r}}f(r,t)}

Thus f ( r , t ) {\displaystyle f(r,t)} uniquely determines the two-point correlation function. Theodore von Kármán and Leslie Howarth derived the evolution equation for f ( r , t ) {\displaystyle f(r,t)} from Navier–Stokes equation as

∂ ∂ t ( u ′ 2 f ) − u ′ 3 r 4 ∂ ∂ r ( r 4 h ) = 2 ν u ′ 2 r 4 ∂ ∂ r ( r 4 ∂ f ∂ r ) {\displaystyle {\frac {\partial }{\partial t}}(u'^{2}f)-{\frac {u'^{3}}{r^{4}}}{\frac {\partial }{\partial r}}(r^{4}h)={\frac {2\nu u'^{2}}{r^{4}}}{\frac {\partial }{\partial r}}\left(r^{4}{\frac {\partial f}{\partial r}}\right)}

where h ( r , t ) {\displaystyle h(r,t)} uniquely determines the triple correlation tensor

S i j = ∂ ∂ r k ( u i ( x , t ) u k ( x , t ) u j ( x + r , t ) ¯ − u i ( x , t ) u k ( x + r , t ) u j ( x + r , t ) ¯ ) . {\displaystyle S_{ij}={}{\frac {\partial }{\partial r_{k}}}\left({\overline {u_{i}(\mathbf {x} ,t)u_{k}(\mathbf {x} ,t)u_{j}(\mathbf {x} +\mathbf {r} ,t)}}-{\overline {u_{i}(\mathbf {x} ,t)u_{k}(\mathbf {x} +\mathbf {r} ,t)u_{j}(\mathbf {x} +\mathbf {r} ,t)}}\right).}

Loitsianskii's invariant

L.G. Loitsianskii derived an integral invariant for the decay of the turbulence by taking the fourth moment of the Kármán–Howarth equation in 1939, i.e.,

∂ ∂ t ( u ′ 2 ∫ 0 ∞ r 4 f   d r ) = [ 2 ν u ′ 2 r 4 ∂ f ∂ r + u ′ 3 r 4 h ] 0 ∞ . {\displaystyle {\frac {\partial }{\partial t}}\left(u'^{2}\int _{0}^{\infty }r^{4}f\ dr\right)=\left_{0}^{\infty }.}

If f ( r ) {\displaystyle f(r)} decays faster than r − 3 {\displaystyle r^{-3}} as r → ∞ {\displaystyle r\rightarrow \infty } and also in this limit, if we assume that r 4 h {\displaystyle r^{4}h} vanishes, we have the quantity,

Λ = u ′ 2 ∫ 0 ∞ r 4 f   d r = c o n s t a n t {\displaystyle \Lambda =u'^{2}\int _{0}^{\infty }r^{4}f\ dr=\mathrm {constant} }

which is invariant. Lev Landau and Evgeny Lifshitz showed that this invariant is equivalent to conservation of angular momentum. However, Ian Proudman and W.H. Reid showed that this invariant does not hold always since lim r → ∞ ( r 4 h ) {\displaystyle \lim _{r\rightarrow \infty }(r^{4}h)} is not in general zero, at least, in the initial period of the decay. In 1967, Philip Saffman showed that this integral depends on the initial conditions and the integral can diverge under certain conditions.

Decay of turbulence

For the viscosity dominated flows, during the decay of turbulence, the Kármán–Howarth equation reduces to a heat equation once the triple correlation tensor is neglected, i.e.,

∂ ∂ t ( u ′ 2 f ) = 2 ν u ′ 2 r 4 ∂ ∂ r ( r 4 ∂ f ∂ r ) . {\displaystyle {\frac {\partial }{\partial t}}(u'^{2}f)={\frac {2\nu u'^{2}}{r^{4}}}{\frac {\partial }{\partial r}}\left(r^{4}{\frac {\partial f}{\partial r}}\right).}

With suitable boundary conditions, the solution to above equation is given by

f ( r , t ) = e − r 2 / 8 ν t , u ′ 2 = c o n s t . × ( ν t ) − 5 / 2 {\displaystyle f(r,t)=e^{-r^{2}/8\nu t},\quad u'^{2}=\mathrm {const.} \times (\nu t)^{-5/2}}

so that,

R i j ( r , t ) ∼ ( ν t ) − 5 / 2 e − r 2 / 8 ν t . {\displaystyle R_{ij}(r,t)\sim (\nu t)^{-5/2}e^{-r^{2}/8\nu t}.}

See also

References

  1. De Karman, T., & Howarth, L. (1938). On the statistical theory of isotropic turbulence. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 164(917), 192–215.
  2. Monin, A. S., & Yaglom, A. M. (2013). Statistical fluid mechanics, volume II: Mechanics of turbulence (Vol. 2). Courier Corporation.
  3. Batchelor, G. K. (1953). The theory of homogeneous turbulence. Cambridge university press.
  4. Panchev, S. (2016). Random Functions and Turbulence: International Series of Monographs in Natural Philosophy (Vol. 32). Elsevier.
  5. Hinze, J. O. (1959). Turbulence, (1975). New York.
  6. Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 36, No. 2, pp. 209–223). Cambridge University Press.
  7. Loitsianskii, L. G. (1939) Einige Grundgesetze einer isotropen turbulenten Strömung. Arbeiten d. Zentr. Aero-Hydrdyn. Inst., 440.
  8. Landau, L. D., & Lifshitz, E. M. (1959). Fluid Mechanics Pergamon. New York, 61.
  9. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics. 1987. Course of Theoretical Physics.
  10. Proudman, I., & Reid, W. H. (1954). On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. R. Soc. Lond. A, 247(926), 163-189.
  11. Batchelor, G. K., & Proudman, I. (1956) The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A, 248(949), 369-405.
  12. Saffman, P. G. (1967). The large-scale structure of homogeneous turbulence. Journal of Fluid Mechanics, 27(3), 581-593.
  13. Spiegel, E. A. (Ed.). (2010). The Theory of Turbulence: Subrahmanyan Chandrasekhar's 1954 Lectures (Vol. 810). Springer.
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