High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex
Highlights
► We compute Prandtl and Navier–Stokes solutions at high Re. ► Large-scale and small-scale interactions within the boundary layer are revealed. ► Large-scale interaction leads to the failure of Prandtl’s BL theory. ► High gradients and splitting of vortex cores characterize the small-scale stage. ► Peaks of enstrophy signal the interaction between dipolar vortex structures.
Introduction
The aim of this paper is to analyze the unsteady separation process of a 2D incompressible Navier–Stokes (NS) flow induced by the interaction of a point vortex with a physical boundary. We shall solve Navier–Stokes equations at different Re regimes (Re = 103–105) and we shall compare these results with the predictions of the classical boundary layer theory (BLT) as expressed by Prandtl’s equations.
Prandtl’s equations can be derived from the Navier–Stokes equations as the formal asymptotic limit for Re → ∞. It is well known how, for many significant flows, Prandtl’s solutions develop a singularity (see for example [14], [4], [23], [24], [13], [12]).
In their seminal work devoted to the analysis of the flow around the impulsively started disk, Van Dommelen and Shen [14] found that, in a finite time, a singularity in the Prandtl solution forms (the VDS singularity). The difficulties that had prevented the previous investigations to give reliable results on the singularity, were solved in [14] using a Lagrangian formulation, which allowed to overcome the problem of the growth in time of the normal velocity component V, i.e. the growth in time of the boundary layer. More recently the same problem has been tackled in [11], [16], [15], where using a high resolution Eulerian spectral method, the authors have tracked the VDS singularity in the complex plane before the real blow up of the solution, and have classified it as a cubic-root singularity.
According to the Moore-Root-Sears (MRS) model the singularity in the solution of Prandtl’s equation is related to the unsteady separation of the boundary layer, see also [25]. In fact the occurrence of a singularity means that the normal component of the velocity V becomes infinite with the relative ejection of vorticity and flow particles from within the boundary layer into the outer flow, with the consequent breakdown of the assumptions which Prandtl’s equation are based on.
Before the occurrence of the singularity, the adverse streamwise pressure gradient imposed across the boundary layer induces the formation of a back-flow region. It has been observed that, generally, the formation of a recirculation region corresponds to the vanishing of the vorticity at a point of the boundary. The back flow region grows in time in the streamwise direction, and ejects farther in the normal direction. This results in the formation of a sharp spike in the displacement thickness and in the streamlines. The spike in the streamlines, at the singularity time, reaches the outer external flow, once again a signal of the interaction of the boundary layer with the outer flow.
As an historical remark we mention that before the important results obtained by Van Dommelen and Shen, the classical definition of unsteady separation was connected with the formation of reversed flow and the vanishing of the wall shear. However Sears and Telionis, in [25], observed that the presence of reversed flow is not in itself sufficient to lead to unsteady separation; they quoted examples of flows with vanishing wall shear for which a breakaway is never expected to occur.
An interesting review on boundary layer theory and on the many numerical experiments which followed Van Dommelen and Shen’s work is given by Cowley in [10]. For the reader interested in the results of the mathematical theory of the Prandtl equations, see [3].
In the rest of the paper we shall denote the Reynolds number with Re defined as:where a and Uc are the distance from the wall and the velocity with respect to the wall of the point vortex, while ν is the kinematic viscosity. Solving the Navier–Stokes equations at different Reynolds numbers we shall follow the unsteady separation process. We shall find significant differences in the behavior of NS solutions between low (103 ⩽ Re ⩽ 2 × 103) and moderate-high (3 × 103 ⩽ Re ⩽ 105) Reynolds number regimes. In fact we shall see how, at different Re, different kinds of interactions will establish between the viscous boundary layer and the outer flow; we shall also see that these interactions are the responsible for the ultimate failure of the Prandtl’s equations to give an accurate approximation of the NS flow at the Re numbers we have tested.
In the classical BLT the streamwise pressure gradient is imposed by the outer flow while the normal pressure gradient is zero to leading order. Therefore through the analysis of topological changes of the pressure gradient, we shall be able to distinguish the different stages of the interaction between the BL and the outer flow. Moreover the evolution of the pressure gradient will give indications on the agreement between the Prandtl and the NS solutions.
During the early stage we shall observe that the Prandtl solution is quite close, both qualitatively and quantitatively, to the Navier–Stokes solutions.
On the other hand relevant discrepancies can be observed when the boundary layer flow starts interacting with the outer flow over a large streamwise scale. This event can be related to the formation of an inflection point in the streamwise pressure gradient. This phenomenon is observed for all Re we have considered.
A second different interaction occurring on a smaller scale is present only for moderate-high Re numbers. We notice that, differently from what happens for lower Re numbers, several local maxima–minima form in the streamwise pressure gradient, forcing the formation of several recirculation regions and of strong gradients in the solution. The first appearance of spiky-behavior in the streamlines and vorticity contour level signals the beginning of this new stage. At this time any comparison with Prandtl’s solution fails, even if the formation of large gradients in the solution resembles the terminal singularity stage of Prandtl’s solution. Both types of interaction (large scale and small scale) begin quite early with respect to the first viscous-inviscid interaction that occurs in Prandtl’s solution.
In the literature there are several attempts to incorporate the interaction between the boundary layer and the outer flow in a theory that would improve the classical BLT. We mention the work in [24], where the authors assume that, as the spike in the displacement thickness grows, the outer flow begins to respond to the boundary layer. However the solutions of these Interactive Boundary Layer Theory terminate with a singularity at a time prior to Prandtl’s singularity time. See also [6] and the discussion in [26]. A possible cure to this was proposed in [27], [20], where the effects of an increasing normal pressure gradient (which is considered constant in the classical BLT) are taken into account. However none of the theories trying to go beyond the classical BLT is fully satisfactory and the problem of a coherent asymptotic theory able to describe the BL separation phenomena is still an open problem, see the discussion in [21] and the review paper [10].
In the next Section we introduce the physical problem, a 2D vortex interacting with a wall, and discuss the regularization procedure we have adopted to treat the relative initial datum. In Section 3 we present the numerical schemes we have used to solve Prandtl’s and Navier–Stokes equations. The numerical results obtained from Prandtl’s equation up to singularity formation are briefly (as this problem was already discussed in great detail in [29], [23]) described in Section 4. In Sections 5 Navier–Stokes results: large-scale interaction, 6 Navier–Stokes results: small-scale interaction, we show the results obtained for the Navier–Stokes solutions, and we analyze the different stages of unsteady separation. In particular the large-scale interaction stage, that develops for all Re numbers we considered, is discussed in Section 5, while the small-scale interaction stage, found for moderate-high is discussed in Section 6. Our analysis follows the treatment of [4], [21], where the authors studied the interaction of the thick core vortex with a boundary and confirms the scenario described in these paper, as well in [17].
In Section 7 we shall discuss in more detail the physical phenomena leading to the different kind of interactions described in the previous Sections; in particular we shall see first the formation of dipolar vortical structures as the signal of the small scale interaction and second a significant increase in the enstrophy of the flow as the result of the movement of these dipolar structures toward the wall. This analysis is influenced by the findings appeared in a recent series of papers [8], [7], [17] and previously in [22], [9], where the case of the collision of a dipole vortex with a boundary was considered.
Section snippets
Statement of the problem
The initial fluid configuration consists of a point-vortex immersed in a 2D viscous incompressible flow at rest at infinity and bounded by an infinite rectilinear wall. The vortex is placed at a distance a from the wall, and is taken with positive rotation and strength k. In the inviscid case, the vortex moves to the right parallel to the wall with constant velocity Uc = k/4aπ. We refer to [18] for more details.
We introduce a cartesian frame (x, y), such that the x-axis coincides with the solid
Numerical schemes for Prandtl’s equations
In this section we explain the numerical method used to solve the boundary-layer Eqs. (2.9), (2.10) with initial and boundary conditions (2.11), (2.12). This problem was first investigated by Walker in [29], and later by Peridier, Smith and Walker in [23]. The problematic numerical instabilities developed by the numerical method of [29] in Eulerian formulation, were overcome using in [23] a Lagrangian formulation, and using an ADI scheme with upwind-downwind differencing approximation for the
Prandtl’s solution
In this section we shall give a description of the physical phenomena occurring in the boundary layer leading to the final break up of the solution due to the blow-up of the first derivative of the streamwise velocity component. In particular we shall describe the various stages leading to separation, and focus our analysis on physical events like the formation of the recirculation region and the first viscous-inviscid interaction.
These phenomena were already discussed in [23], [24] and
Navier–Stokes results: large-scale interaction
In this section we shall study the behavior of the solutions of the Navier–Stokes equations at different Re numbers (103–105). We shall also be interested in the comparison between the Navier–Stokes solutions and Prandtl’s solution up to the singularity time ts = 0.989. In particular we shall investigate the interaction between the viscous boundary layer and the inviscid outer flow occurring during the various stages of the unsteady boundary layer separation.
Before the beginning of this
Navier–Stokes results: small-scale interaction
The characteristics of the large-scale interaction bear no resemblance with the viscous-inviscid interaction developed by Prandtl’s solution which is characterized by the formation of a spike in the streamlines and vorticity contours. However the large-scale interaction in Navier–Stokes solutions is the precursor of another interaction, acting on a smaller scale. We shall see that this phenomenon occurs only for moderate-high Re (i.e. 104 ⩽ Re ⩽ 105) numbers.
Separation, dipolar structures and vorticity production
The description of the unsteady separation of the previous sections was based on the analysis of the evolution of the streamwise pressure gradient and of the vorticity at the wall. In this Section we shall look at the boundary layer dynamics from a different perspective. Namely we shall see how, for moderate-high Re, an important event occurring during the separation process, is the creation of several vortex-dipoles, and that the reciprocal interaction between these structures leads to a sharp
Conclusions
We have computed the solutions of 2D Prandtl and Navier–Stokes equations in the case of a rectilinear vortex interacting with a wall. We have analyzed the asymptotic validity of boundary layer theory by comparing Prandtl’s solution with the NS solutions for Re in the range 103–105. In our case Prandtl solution terminates in a singularity at time t ≈ 0.989. The singularity formation is anticipated by a first interaction of the boundary layer flow with the outer flow, which is revealed by the spiky
Acknowledgements
The authors thank Kevin Cassel for several interesting discussions on the topics of the present paper and his helpful comments. The authors also thank the reviewers for comments and suggestions that significantly helped in improving the paper. The present work has been supported by INDAM and by the Math Dept of the University of Palermo through the FPR grant.
References (30)
- et al.
Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations
Appl Numer Math
(1997) - et al.
The normal and oblique collision of a dipole with a no-slip boundary
Comput Fluids
(2006) - et al.
Fundamental interactions of vortical structures with boundary layers in two-dimensional flows
Physica D
(1991) - et al.
The spontaneous generation of the singularity in a separating laminar boundary layer
J Comput Phys
(1980) - et al.
Singularity formation for Prandtl’s equations
Physica D
(2009) - et al.
Computational fluid mechanics and heat transfer, Series in computational methods in mechanics and thermal sciences
(1984) - et al.
Existence and singularities for the Prandtl boundary layer equations
ZAMM Z Angew Math Mech
(2000) A comparison of Navier–Stokes solutions with the theoretical description of unsteady separation
Philos Trans Roy Soc Lond A
(2000)- et al.
instability in a vortex-induced unsteady boundary layer
Phys Scripta
(2010) - et al.
The effect of interaction on the boundary layer induced by a convected rectilinear vortex
J Fluid Mech
(1989)
Dissipation of kinetic energy in two-dimensional bounded flows
Phys Rev E
Singularity tracking for Camassa–Holm and Prandtl’s equations
Appl Numer Math
The boundary layer induced by a convected two-dimensianal vortex
J Fluid Mech
On the Lagrangian description of unsteady boundary-layer separation. I. General theory
J Fluid Mech
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