Nonlinear structures of transition in wall-bounded flows - Science Direct

Applied Numerical North-Holland

Mathematics

7 (1991) 129-150

129

Nonlinear structures of transition in wall-bounded flows S. Biringen Department

and

E. Laurien

*

of Aerospace Engineering

Sciences, University of Colorado, Boulder, CO 80309, USA

Abstract Biringen, S. and E. Laurien, Nonlinear Mathematics 7 (1991) 129-150.

structures

of transition

in wall-bounded

flows,

Applied

Numerical

We review recent developments on the numerical simulation of flow structures evolving during the nonlinear stages of transition to turbulence in wall-bounded flows. Plane Poiseuille flow and the Blasius boundary layer are considered as model problems for which simulations have been performed by solving the three-dimensional time-dependent incompressible Navier-Stokes equations in a spatially periodic integration domain. Numerical techniques including finite-difference, spectral and pseudospectral methods used in these simulations are also summarized. The transition process considered here is a nonlinear phenomenon which arises from the amplification of initially small-amplitude two- and three-dimensional disturbances. During the later stages, nonlinear velocityvorticity field interactions instigate the evolution of characteristic three-dimensional structures such as lambda vortices, horseshoe vortices and hairpin vortices. Significant advances have recently been made in both experimental and numerical studies of these structures. We give an account of the numerical work and compare with experiments.

1. Introduction 1.1. The physical problem Transition from laminar to turbulent flow has been investigated theoretically and experimentally for many decades. In the laminar state, particles in a fluid move almost parallel to each other, pathlines and streamlines are relatively smooth and regular. In contrast, a turbulent flow has an unsteady three-dimensional and seemingly unordered character. Most flows, both in na,.ure and in technological applications, are turbulent or are in a state of transition to turbulence. In this article, we focus on the transition process as it occurs in flows along rigid walls. Our understanding of transition can provide an insight to the basic mechanisms of turbulence. Furthermore, there is substantial technical interest to describe, control and model transition in order to increase efficiency of turbomachines, airplane wings and hulls. In the last two decades, significant progress has been made in the investigation of transition in plane Poiseuille flow and the Blasius boundary layer for which the laminar states are well known. * Present address: German Aerospace Research Giittingen, Germany; visiting at the University 0168-9274/91/$03.50

0 1991 - Elsevier

Establishment of Colorado

Science Publishers

(DFVLR), 1987/88.

Institut

B.V. (North-Holland)

for Theoretical

Fluid Mechanics,

130

S. Biringen, E. Laurien / Nonlinear structures of transition

We consider these flows as our model problems. In boundary layers, the mean velocity is two-dimensional, i.e., only a function of two spatial coordinates, which we denote with x (downstream) and z (normal to the wall), and there is no average velocity in the spanwise direction y (normal to the x-z plane). The plane Poiseuille flow is the flow in a channel between two parallel plates driven by a downstream pressure gradient and the mean velocity is a function only of z. The Blasius boundary layer is the flow along the surface of a flat plate with zero pressure gradient driven by an outer parallel flow. In these flows, transition to turbulence can occur via many routes, depending on wall roughness, disturbance environment, temperature, and vibration; for a detailed discussion of this issue, we refer to the articles by Tani [57] and Herbert and Morkovin [24]. In order to isolate these factors, most experimental research has been done in low turbulence wind or water channels with well-defined small initial disturbances which amplify and develop as they travel downstream from the disturbance source. In many experimental studies, transition has been initiated by a ribbon or wire vibrating with a given constant frequency, which can be controlled. We will refer to those experiments, summarized e.g. by Arnal [2], below. In these experiments, transition develops in a very regular fashion, which is periodic in time and enables accurate measurements of the flow field variables. In both of our example flows, similar characteristic structures have been observed downstream of the vibrating ribbon. These were clearly identified and found to be reproducible in the classical experiments of Klebanoff, Tidstrom and Sargent [25], Hama and Nutant [20] and Kovasznay, Komoda and Vasudeva [30] in the Blasius boundary layer and Nishioka, Asai and Iida [44] in plane Poiseuille flow. Before the advent of high-speed computers a theoretical explanation or analytical description of these structures could only be given by the linear theory for the initial stages of flow instability. In this theory, the governing equations of motion are linearized due to the assumption that the disturbances are small and the amplitude, phase distribution as well as wave speed and amplification of two-dimensional Tolmien-Schlichting waves or three-dimensional oblique waves can be predicted; detailed treatments of this subject are available in Betchov and Criminale [4], Drazin and Reid [12], Mack [35], and Nayfeh [41]. In a certain Reynolds number and frequency range, the Tollmien-Schlichting waves become their amplitude grows as they travel downstream. The development of such unstable, i.e., disturbances instigate three-dimensionality which, in turn, promotes interactions between the velocity and vorticity fields leading to strong nonlinear effects. Consequently, at this stage the above assumption of small disturbances and linearity is not valid. In this paper, we review recent progress in the simulation and description of the nonlinear structures in wall-bounded flows. This progress has become possible with the recent advances in applied numerical analysis and in expanding computer resources. Efficient numerical techniques have made numerical simulations of nonlinear structures possible on supercomputers. 1.2. Linear stability theory To describe small-amplitude disturbances, linear stability theory has been very successfully employed. In this theory, all disturbances are assumed to have an infinitesimally small amplitude, and consequently the equations of motion can be linearized. Assuming that disturbances are strongly periodic (harmonic) in time and space and that they travel downstream with a

S. Biringen, E. Laurien / Nonlinear structures of transition

constant

velocity, they can be expressed * = Re{ @(z> e[i(*-~+Pv)-iacrl}.

by a stream function,

131

‘k, of the form of a traveling

wave: (14

This wave form contains a complex amplitude function Q(z), which expresses the wave amplitude and phase as a function of the wall-normal coordinate z, the wave numbers (Y and p and the wave velocity c. Using this wave form, the stability problem can be reduced to an ordinary differential equation for the amplitude function which is known as the Orr-Sommerfeld equation:

(u - c)( @” - cl*@)- U”@ + -

ciRe(@

“”

-

2(3@”

+ &p)

=

0,

0.2)

where a prime denotes differentiation with respect to z, i = m and U = U(z) corresponding to the velocity profile of the basic flow. In the case of a boundary layer, the use of this equation implies a constant local Reynolds number Re and the exclusion of nonparallel effects. Both assumptions can only be justified by the good agreement of the results with experimental observations and will not be discussed here. From the no-slip condition, the following boundary conditions for @ are obtained, @(_tl)

=@‘(,l)

in the case of Poiseuille

=o

(1.3)

flow, and

Q(O) = @(z + cc) = 0,

(1.4a)

Q’(O) = @‘(z -+ cc) = 0

(1.4b)

in the case of the boundary-layer flow. Mathematically the problem is an eigenvalue problem for @. A linear stability analysis can formally be done in two different ways: the first is to define (Y= (Y,+ icui to be complex and c to be real in (1.1). This is referred to as the spatial analysis where disturbances are periodic in time and develop spatially in x. The second, the so-called temporal analysis, assumes (Y to be real and c = c, + ic, to be complex. Here, disturbances are periodic in space and develop in time. Physically, a vibrating ribbon experiment is a space-evolving problem and the spatial analysis is appropriate. In this case, c = a/(~, and w, the real vibration frequency of the ribbon, is given; (Yis the complex eigenvalue of the problem. Due to the nonlinear appearance of the eigenvalue in (1.2), the spatial problem is more difficult to solve than the temporal case which consists of a linear eigenvalue problem. A spectral method to solve the spatial (nonlinear) eigenvalue problem is given by Danabasoglu and Biringen [66]. The temporal approach has also been considered to be more feasible for the full simulation of the nonlinear problem since it allows the use of the available computing resources to resolve only one wave length of the disturbance in a reference frame travelling with the phase speed of the disturbance wave. This is the approach adopted in the problems that will be considered in this paper. 1.3. Numerical

simulation

model

The equations of motion governing the full transition time-dependent incompressible Navier-Stokes equations:

a

%u+u.v

‘u=

-p

1

vp+s

1

v*u,

problem

are the three-dimensional

(1.5)

132

S. Biringen, E. Laurien / Nonlinear structures of transition

where u is the velocity vector, p is the pressure, Re = UrjrefIref/v is the Reynolds number, and v is the kinematic viscosity of the fluid. The definition of the Reynolds number depends on the flow geometry. In the case of Poiseuille flow, the reference velocity Uref is the mean flow velocity at the center line and the reference length lref is the half-width of the channel. For the Blasius boundary layer, Uref is the velocity of the outer flow and lref the local displacement thickness or the thickness of the boundary layer. The second-order term on the right-hand side of (1.5) represents momentum transport by diffusion, and the nonlinear terms on the left-hand side describe momentum transport by convection. On the rigth-hand side, there is also the pressure gradient term. The condition of mass conservation for an incompressible fluid leads to the continuity equation V .u=o,

(1.6)

which has to be satisfied at all times. Boundary conditions for the system of equations Zl=O

to be solved are the no-slip

condition (1.7)

at rigid walls, and the far-field

condition

II = [l, 0, o]

(1.8)

as z -+ cc in the case of the boundary layer. Further, velocity boundary conditions are needed at the upstream, downstream, and side boundaries of the integration domain. In most experiments it has been observed that the transition process is almost periodic in the spanwise direction, so that the characteristic flow structures repeat themselves within a certain spanwise wave length. Therefore, in numerical simulations the boundary condition implying spanwise periodicity u(x,

Y, z, t) =+,

Y f L,,

z, t>

(1.9)

is well justified. Here, the spanwise wave length L,, often expressed as a wave number CX~ = 2a/L,, is a given parameter. When a certain experiment is simulated, the value of (Ye may be obtained from the experimental data. Another very important assumption, which has been used in most of the numerical results considered here, is the periodicity in the downstream direction which implies: u(x,

Y, 2, t) =u(x+L,,

Y, 2, t>.

(1 .lO)

This assumption is an extension of the temporal approach into the nonlinear regime. The spatial periodicity of the integration domain allows the application of efficient Fourier spectral methods in the periodic directions, to which we will refer in the next section. The temporal simulation model has first been used by Orszag and Kells [48] in Poiseuille flow and was later shown to be very accurate up to the spike stage by Kleiser [26,27], who provided extensive comparisons of his results with the experiments of Nishioka [44]. Qualitative agreement with Nishioka’s results were further furnished by the simulations of Biringen [5], Biringen and Maestrello [S], and Zang, Krist, Erlebacher and Hussaini [65] albeit the Reynolds number difference between the experiment and these simulations. In the boundary layer, additional assumptions are necessary to apply to the temporal model such as the parallel flow assumption.

t L

13 a)

Fig.

1. Rms

0%

t

*?

x3

5

5

5

10 0

t

133

S. Biringen, E. Laurien / Nonlinear structures of transition

0

b)

0 000

10 0

0.10"',_0.15

fluctuations (u;) along numerical simulation

CJ

5

the wall-normal coordinate in the Blasius [34] with experiment [60]; -, simulation,

+

10 a 0

0.05

boundary layer. o, experiment.

0.10.;_0.15

Comparison

of

Despite this somewhat less obvious assumption, the studies of Wray and Hussaini [62], Spalart [55], Laurien and Kleiser [34], and Laurien [32] provide good qualitative agreement with experiments (Fig. 1). Therefore, the temporal model seems to be justified for the simulation of the mechanisms and flow structures in the nonlinear stages of transition in both the plane Poiseuille flow and the Blasius boundary layer. Quantitative predictions, especially in the late, highly nonlinear stages of boundary-layer transition, however, may lack accuracy with the temporal model. Here, simulations with the spatial approach should be expected to yield better results [14,29,40,67].

2. Numerical techniques to solve the governing equations 2.1. Temporal and spatial discretization The physical problem described in the previous section has been solved by various numerical algorithms, which will be discussed in this section. In transition problems, important criteria for the selection of an algorithm are the accuracy and the number of floating-point operations as well as computer storage needed to achieve this accuracy. These criteria follow from the three-dimensional nature of the problem and from the requirement to perform a time-accurate numerical simulation. We consider the numerical integration of equations (1.5)-(1.6) with the boundary conditions (1.7) and (1.8). Discretization in time in most algorithms is done using mixed implicit-explicit methods. The linear viscous term is usually treated implicitly by the Crank-Nicolson scheme and the convective terms are advanced by the explicit Adams-Bashforth or Runge-Kutta methods. These discretizations typically are second-order accurate in time. Along the streamwise x and lateral y directions, the flow is assumed to be periodic and flow field variables are expanded in terms of finite Fourier series. e.g., u (or ui) can be written as: ui(x,, n,=

z, x3) = C C&(k,, “I “i -:NJ

)...) O,l)...)

z, k3) exp[i(k,x,

+ k,x,)]. (2.1)

:N,-1.

In expressions (2.1) k, = 2nn,h,/N,, h, is the grid spacing in the x, direction, whereas N, is the number of mesh points in the x, direction. For the linear term this spectral method is very efficient, since the three-dimensional problem breaks down into iN,N, complex one-dimensional problems in the coordinate z (or N,), which

134

S. Biringen, E. Laurien / Nonlinear structures of transition

are completely decoupled. Consequently, the decoupled equations can be solved for fi, and the variables in physical space are obtained by a back Fourier transform. The theory of spectral methods has been covered in various books [17,58]; for a recent overview see [9]. The treatment of the nonlinear terms in spectral space, however, may be very inefficient, because products of the double sum (2.1) have to be computed. The term udu/dx, for instance, would require a total number of 0( N:&*) floating-point operations. A much more efficient method is to use the pseudospectral method [46] to obtain spatial derivatives and calculate the product in physical space. When Fast Fourier Transform (FFT) algorithms are used, each Fourier transform requires only 0( Ni log N,N, log N3) operations. The computation of the nonlinear terms requires 0( NiN,) operations. This method is clearly superior to the treatment in real space when Ni and N3 are large (e.g. about 50). In most cases, the convective terms are written in the rotational form that prevents occurrence of nonlinear instability by ensuring conservation of momentum and energy. In vector form, this can be written as u. vu

= u X rot(u)

- vu’.

(2.2)

The second term on the right-hand side can be included into the pressure gradient defining a new variable 4 = p + u*. The mechanism of energy transfer from low frequency modes (small k,, k3) to higher frequencies (large k,, k3) can easily be explained in terms of the Fourier expansions. Initially, only a few low frequency modes are excited. Because of the quadratic nature of the nonlinear terms in the Navier-Stokes equations, energy may be transferred to the modes corresponding to sums of wave number combinations in the exponents of (2.1) at each time step. Thus increasingly higher frequency modes become excited in the course of the transition process. Due to the truncation of the Fourier series, a certain type of numerical error may occur as follows [46]: the energy can only be transferred to higher frequency modes as long as those modes are present in the discretization. When the truncation limit is reached, the transfer mechanisms become nonphysical and lead to misinterpretation of higher frequencies generated during the computation of the nonlinear terms in real space as low frequencies due to the periodicity of the Fourier expansions. These errors are called “aliasing errors” and occur when the resolution of the pseudospectral method becomes too low to represent the complete spectrum generated during the computation. Aliasing errors can be reduced [46] by introducing additional high frequency modes, which are only used during the Fourier transforms. Thus high frequencies will be neglected rather than misinterpreted. These techniques are, of course, only applicable as long as neglected frequency amplitudes are small. However, the work of Krist and Zang [31] on the simulation of transition in plane Poiseuille flow indicates that both aliased and de-aliased calculations remain valid until they lose resolution and the de-aliased calculations retain resolution only slightly longer than the aliased calculations. The solution of the equation along the third direction can be obtained in several ways. In an algorithm devised by Moin, Reynolds and Ferziger [38], the continuity equation is solved directly along with the momentum equations. When the governing equations are Fourier transformed, the problem reduces to a system of four coupled ordinary differential equations which must be along z-direction can be done by either finite solved for every pair of k,, k,. Discretization differences or by Chebychev approximations. This method was applied to transition in plane Poiseuille flow by Biringen [5]. The algorithm of Kleiser and Schuman [28] requires the solution

S. Biringen, E. Laurien / Nonlinear structures of transition

of one-dimensional u”(z) for a variable the Poiseuille

Helmholtz

equations

of the general

135

form

- b2U(Z) = r(z)

(2.3)

U(Z); the equation is solved with Dirichlet flow, the integration interval is

or Neumann

boundary

conditions.

-lBz