Steady-state Burgers turbulence with large-scale forcing

• Published in: Physics of fluids (1994) 1998-11, Vol.10 (11), p.2859-2866
• Main Authors: Gotoh, Toshiyuki, Kraichnan, Robert H.
• Format: Article
• Language: English
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.869807

Steady-state Burgers turbulence supported by white-in-time random forcing at low wave numbers is studied analytically and by computer simulation. The peak of the probability distribution function (pdf) Q(ξ) of velocity gradient ξ is at ξ=O(ξ f ), where ξ f is a forcing parameter. It is concluded that Q(ξ) displays four asymptotic regimes at Reynolds number R≫1: (A) Q(ξ)∼ξ f −2 ξ exp (−ξ 3 /3ξ f 3 ) for ξ≫ξ f (reduction of large positive ξ by stretching); (B) Q(ξ)∼ξ f 2 |ξ| −3 for ξ f ≪−ξ≪R 1/2 ξ f (transient inviscid steepening of negative ξ); (C) Q(ξ)∼|Rξ| −1 for R 1/2 ξ f ≪−ξ≪Rξ f (shoulders of mature shocks); (D) very rapid decay of Q for −ξ⩾O(Rξ f ) (interior of mature shocks). The typical shock width is O(1/Rk f ). If R −1/2 ≫rk f ≫R −1 , the pdf of velocity difference across an interval r is found to be P(Δu,r)∝r −1 Q(Δu/r) throughout regimes A and B and into the middle of C.