A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows
Introduction
Direct numerical simulations (DNS) and large eddy simulations (LES) of incompressible flows are generally attempted using either primitive or derived variable formulations. Fractional step methods with three time level Adams–Bashforth scheme for time integration are generally used in primitive variable formulation [1] for time-accurate solution of Navier–Stokes equation (NSE). It has been shown in [2] that use of three time level Adams–Bashforth algorithm introduces spurious numerical mode, which removes high frequency components of the initial solution, invalidating any claim of performing DNS by this approach. Another difficulty with this approach is the specification of wall boundary conditions at the intermediate stage. This is usually achieved via extrapolation from values obtained in previous time steps. Perot [3] has shown that while this fix may provide numerical stability, but it gives rise to large time splitting error. To minimize this error, one is again forced to take small time steps. An approach that avoids this problem by developing correct conditions is given in [4], but is very memory intensive [5]. In [6], fourth-order accurate explicit and compact finite-difference methods for incompressible NSE in -formulation is presented where four-stage, fourth-order Runge–Kutta () time integration scheme is used. In this formulation, the solenoidality constraint on velocity is enforced for each Runge–Kutta stage by solving respective Poisson equations for pressure correction and is computationally very intensive.
Thus, it is natural to seek formulations which eliminate pressure altogether from the governing equations. In 2D flows, we note the advantage of stream function-vorticity -formulation over the primitive variable formulation, for reasons of lesser number of unknowns as compared to three variables in -formulation; exact satisfaction of mass conservation everywhere in the flow field and obtaining vorticity directly from the governing equation and not by numerical differentiation of computed quantities. However, this formulation's extension to 3D flows by using vector potential-vorticity formulation is not straightforward due to complex and coupled boundary conditions. In view of this, -formulation is more amenable, as we show the equivalent governing equation for in -formulation to be entirely different, which allows for a simple satisfaction of condition, as compared to -formulation.
-formulations have been used by several researchers to solve external [7] and internal flow problems [8]. Different references report different forms of -formulation. Mostly, two forms of this formulation have been used: (i) a non-conservative form [9], [10], [11] and (i) the Laplacian form [8], [12]. The difference between these two formulations lie in the way nonlinear convective and vortex-stretching terms in the vorticity transport equations are handled. In non-conservative form, the vorticity transport equation (VTE) is given as
Note that the first term on the right hand side of Eq. (1) is the vortex stretching term, which is absent for 2D flows. In contrast, the Laplacian form of VTE is given as
Some of the references on -formulations mentioned above, also discuss about the solenoidality conditions on velocity and vorticity. For example, Laplacian form of -formulation is used in [13], where the solenoidality condition on vorticity is enforced based on evolution equation for divergence of vorticity (). In this reference, the authors also claim that their formulation implicitly satisfies solenoidality condition on velocity due to the way the governing equations are derived. Laplacian form has also been used in [12], [14]. In [14] solenoidality of vorticity is satisfied by integrating the wall-normal component of vorticity () from free-stream to the wall for flows over a rotating circular disk for which no evolution equation is solved. However, such procedure is only applicable for the external flows for which the component of vorticities at the free-stream are zero. In [12], no such attempt is made, although the computations are claimed to exactly satisfy divergence-free conditions for both velocity and vorticity. Non-conservative forms of the -formulation are used in [10], [11], [12]. Evolution equation for is reported in [8], where schemes are constructed to satisfy the solenoidality condition on . In [15], the computed vorticity field ω is made divergence-free by adding to it the gradient of a scalar function as where is the unknown divergence-free vorticity vector. Taking divergence of Eq. (3), one obtains the governing equation for the scalar as Hence in [15], four Poisson equations are solved at each time-step, which makes the procedure computationally more expensive and time consuming. While adopting such procedure of obtaining divergence-free vorticity field, one encounters the problem of specifying the boundary condition for which is addressed in [8]. However, all the 3D LDC results reported in [8], [12], [15], [16] are only for the subcritical Reynolds number for which divergence error in vorticity does not affect the final flow structures. This aspect is also shown in the present work via results obtained for the cubical LDC.
In the present work, we report a new rotational form of -formulation in conservative form. We have also derived the evolution equations for divergence of vorticity for the rotational, Laplacian and non-conservative forms of this formulation and shown that enforcing solenoidality condition on vorticity is much simpler for rotational form compared to the other two forms. In the present work, we have used optimized compact schemes for spatial discretization, which have been shown in [17] to have superior spectral accuracy compared to traditional schemes and specially central difference schemes. Also, an optimized Runge–Kutta scheme [18] is used for time integration which provides high accuracy solution to the problems solved here. This formulation is used to solve flow inside 3D rectangular LDC comparing the obtained results with experimental results reported in [19]. Results obtained for cubical LDC are also used to compare the Laplacian and rotational forms. The method used in [14] to solve the disturbance evolution for flows over a rotating disc is applicable only for external flows, as Eqs. (12) to (14) in this reference can be applied to external flows only. On the other hand, our method reported in the present work is general and can be applied to both external and internal flows.
DNS of 3D NSE for the receptivity of flows past a flat plate using Laplacian form of -formulation are also reported in [7], to simulate the evolution of disturbances in 3D flow field. However, this references does not comment about the effects or satisfaction of solenoidality conditions on vorticity. The enforcement of wall-boundary condition on vorticity in [7] is not obtained in the conventional way of specifying as given by Eqs. (12a)–(12c) in this reference.
The other test example considered in the present work involves simulation of transition of an equilibrium boundary layer excited by a Gaussian bump and spanwise modulated wall-exciter in the 3D framework by DNS using -formulation in rotational form. Corresponding DNS of 2D Navier–Stokes equation have been reported in [20], [21]. A numerical investigation of velocity and vorticity field in boundary layer transition process are reported in [22]. In [23], the boundary layer was shown to support growing spatio-temporal wave fronts, which have been shown to grow and cause transition to turbulence in [20]. The deterministic disturbances were shown from the receptivity stage to the fully developed turbulent stage by solving the 2D Navier–Stokes equation without any limiting assumptions. An attempt is made to see that if this is also the case for 3D transitional flow. Solution of such flow in 3D framework has been reported in [24].
The governing equations for -formulation and the evolution equations for and have been developed in the next section. Computational details for: (i) flow inside 3D rectangular LDC and (ii) 3D transition of an equilibrium boundary layer over a flat plate caused by a Gaussian bump and spanwise modulated exciter have been provided in this section. In Section 3, results and discussion for these two test cases have been provided. The paper closes with summary and conclusions in Section 4.
Section snippets
Governing equations and numerical formulation
The governing equations for rotational form of VTE are described next. Additionally, the associated velocity Poisson equations are also derived. In the following, numerical methods adopted to solve the problem are discussed in details.
Computational details, boundary conditions, results and discussion
To show the efficacy of the proposed rotational formulation, using the high accuracy compact schemes of interpolation and derivative evaluation, here we have studied test problems, which simulate flows from onset to transitional and turbulent states exhibiting typical power spectrum.
Summary and conclusions
Here, we have derived a new -formulation of Navier–Stokes equation in rotational form of the VTE. This formulation allows simpler satisfaction of the solenoidality condition on vorticity, which makes it superior to the traditionally used primitive variable formulations or the Laplacian and the non-conservative form of the -formulation. In all the computations reported, solenoidality of the velocity is also preserved by exactly satisfying the divergence-free condition on velocity.
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