Modulated stagnation-point flow and steady streaming

Abstract

Plane stagnation-point flow is modulated in the free stream so that the velocity components are proportional to KH + K cosωt. Similarity solutions of the Navier-Stokes equations are examined using high-frequency asymptotics for K and KH of unit order. Special attention is focused on the steady streaming generated in this flow with strongly non-parallel streamlines. For small modulation amplitude K [les ] KH, unique self-similar streaming flows exist. For large modulation amplitude K > KH, if (K/ω) (K/KH) [ges ] 1.661 no self-similar streaming is possible, while if 4/3 [les ] (K/ω) (K/KH) [les ] 1.661, then multiple steady solutions occur.

Stagnation Point Flow and Heat Transfer Under Free-Stream Turbulence

“…Similarly modulated stagnation point flows were also studied by Ishigaki (1970) and Pedley (1972), but with an emphasis on the surface skin friction. Merchant and Davis (1989) examined the steady streaming driven by large amplitude and high frequency temporal free stream modulation. The results show that the streaming, when it exists, is always confined within the Hiemenz boundary layer thickness.…”

“…Theoretically, the efi-ect of temporal modulation of free-stream velocity was first studied by Lighthill (1954) who obtained the Stokes-Iayer corrections for skin friction and heat transfer for a twc^dimensional pulsating mean flow about a cylinder^ The steady streaming (second order alteration to 7e mean flow owing to the Reynolds stresses) in an oscillatory Hiemenz bounded DaJis^aJsqlh rr'""" ^" ""'"'' "^' '^"^" (^^^2) ^"^ M-^^-t and Davis (1989), but the emphasis is on finding similarity solutions and Itrt K^ « " u^'" ^"'*'°"-^^" enhancement of heat transfer in a perturbed Hiemenz boundary layer was also conceived as a consequence of mod^gymT ^'^J'i' *'' '"'""'"^ disturbances. However, Kestin and Wood (1970) found, and later much clarified by Wilson and Gladwell (1978) that the two-dimensional Hiemenz boundary layer is always linearly stable to the mcoming three dimensional disturbances.…”

Oscillatory Flow Near a Stagnation Point

“…While they were not able to determine the radius of convergence of their series exactly, they estimated that for large σ the series should converge when ∆ < σ. Later Merchant and Davis [7] showed that if both the amplitude and the Strouhal number are large, and if the thickness of the main boundary layer and the induced steady-streaming layer are chosen to coincide, then no solutions exist in our notation when ∆ > 1.289 σ 1/2 . This derives from the fact that the leading order steady-streaming equation has no solution when the amplitude exceeds this bound.…”

“…Previously Grosch and Salwen [6] investigated this problem for small fluctuation amplitudes in the low and high frequency limits. Merchant and Davis [7] also considered the large amplitude, high frequency limits, establishing an asymptotic structure in which the streaming region above the Stokes layer is the same thickness as the Hiemenz boundary layer. We have studied the same flow and, for general parameter values, provided numerical evidence that for all frequencies there exists a threshold value of the amplitude beyond which the flow will break down in finite time.…”

“…Other studies pertinent to this modified Hiemenz problem include those by Grosch and Salwen [6] and Merchant and Davis [7]; among earlier investigations, we mention those by Matunobu [8], Pedley [9], and Ishigaki [10]; see also Lighthill [11]. Grosch and Salwen, while confining their attention to small free stream fluctuations, examined the flow when the frequency of these disturbances is either small or large.…”