The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

Abstract

In this paper a model for viscous boundary and shear layers in three-dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid. The mechanism of entrainment is represented by a discontinuity in the normal component of velocity across the sheet. The sheet mass is able to support a pressure jump, which in turn may cause additional entrainment. This feature was confirmed when the model was used to represent the Falkner-Skan boundary layers. The velocity field induced by the vortex-entrianment sheet is given by a generalized Birkhoff-Rott equation with a complex sheet strength. The model was also applied to the case of separation at a sharp edge. There is no requirement for an explicit Kutta condition in the form of a singularity removal as this condition is inherently satisfied through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model was demonstrated on two shedding example problems. arXiv:1801.07346v2 [physics.flu-dyn]

New insights from inviscid modelling of starting flow separation with roll-up

“…Hence, for this problem set-up, there is a respective efflux and influx of fluid across the boundary at infinity since shares the same sign. A similar phenomenon occurs for Falkner–Skan boundary layers, whereby the vertical flow at infinity changes from an efflux to an influx as the outer flow acceleration is increased, which competes with the displacement of the layer growth due to diffusion (see DeVoria & Mohseni 2019). Motivated by this, we label as ‘diffusion-dominated entrainment’ (DDE) and as ‘inertia-dominated entrainment’ (IDE).…”

“…In two dimensions, the velocity induced by the sheet at an arbitrary point in an otherwise unbounded domain can be expressed by the following generalized Birkhoff–Rott equation, i.e. Cauchy-type integral (DeVoria & Mohseni 2019): The integration is over the sheet whose position at time is with as the arclength coordinate. In general, the dynamics of the sheet is governed by ‘surface equations’ resembling the Euler equations with a sheet density (mass per unit area) and with additional source terms representing the entrainment/flux of mass and momentum.…”

“…As such, the stress on the surface side can be evaluated and integrated to give an estimate of the force on the body, which includes a viscous effect owing to the entrainment. For more details, the interested reader is referred to DeVoria & Mohseni (2019).…”

“…Inviscid models are useful tools for prediction as well as cheaper alternatives to fully resolved simulations. Recently, the authors developed an inviscid model for viscous shear and boundary layers termed a vortex-entrainment sheet (DeVoria & Mohseni 2019). This irregular surface is the superposition of a conventional vortex sheet having a discontinuity in the tangential velocity (i.e.…”

A theoretical model for the separated flow around an accelerating flat plate using time-dependent self similarity

“…3,11). In DeVoria & Mohseni [17] we significantly augmented the solution space by considering an entrainment boundary condition (in place of no through-flow) that allows a complex coefficient A 1 ∈ C and the shedding of a vortex-entrainment sheet [19].…”