Numerical Modelling of Viscous Flows at Low Reynolds Numbers

M. KHALID, W. YUAN, M. MAMOU and H. XUInstitute for Aerospace Research (IAR)

National Research Council (NRC), CANADA

Abstract: - Low Reynolds number flows have attracted renewed attention owing to their complex features. This is particularly true in transitioning turbulent regimes, where enormous effort is required to address the appropriate grid density, orthogonality and time step size to capture the flow physics with its true characteristics. At IAR/NRC, we have developed specialized computational fluid dynamics (CFD) codes that address low Reynolds and Mach numbers and incompressibility issues using appropriate coupling with both one-equation and more rigorous two-equation models. Such low Reynolds number incompressible flows contain additional challenges for small sized vehicles with moving parts representing the motion of flapping, oscillating and rotating wings. Here, tiny vortical structures created by slowly moving wings or other unsteady flow separation effects present enormous grid management and flow resolution problems. Elsewhere, we have experimented with both large eddy simulation (LES) and direct numerical simulation (DNS) to capture sub-grid level transient scales. The codes developed at IAR were first used to investigate, amongst other viscous quantities, the anisotropic turbulent Reynolds stresses in concave and convex corners of a square annular duct as well as in a confined square co-axial jet. These codes have now been updated and applied towards more curvilinear configurations. The prediction of transition to turbulence is another important area of research, and a numerical technique for solving the Orr-Sommerfeld equations has evolved to study stability and transition phenomena in different laminar fluid flow problems. The stability analysis is carried out by imposing infinitesimal perturbations to the basic laminar flow and subsequent discretization of the spatial and temporal growths in the Orr-Sommerfeld equation. This technique is used to determine the thresholds above which the steady laminar flows become unstable to small perturbations.

Key words: - Incompressible flows, low Reynolds number effects, LES, DNS, transition modelling, unsteady flows

1 Introduction

With increasing emphasis on the miniaturization of vehicles for UAV-type applications, there has been a revival of research into flows at low Reynolds numbers. For full-scale aircraft, there were always regions near the nose of the aircraft or leading edges of wings and other control surfaces where the low Reynolds number flow peculiarities contributed to the subsequent characteristics of the flow field. However, in practice, the global performance was mostly driven by the larger Reynolds number effects encountered in most of the flow surrounding the vehicle. A closer scrutiny of the viscous sub-layer testified to the aerodynamic influence of the lower Reynolds number regimes. Indeed, specialized turbulence models were designed to address their dominance near the wall, but it was the outer transitional and fully turbulent layer that contained the bulk of the kinetic energy to sustain turbulence.

This research becomes even more challenging when one considers smaller insect size microvehicles moving at very low speeds, causing incompressibility transients that add to the complexity of the low Reynolds number effects pervading across the entire flow field. What complicates the picture is the behaviour of the viscous subscale transients, which require relatively longer dwell times to evolve and mature at low Reynolds numbers. These in turn breakdown or cascade into still smaller structures. At higher Reynolds numbers, the larger mean velocity dampens their growth and sweeps them continuously towards the outer flow. Tennekes and Lumley [1] emphasize that mean velocity driven dissipation effects stunt the growth of small-sized structures and erode the vorticity development. Launder [2] has defined the local Reynolds number based on the fluctuating velocity for such regions to be of the order of k2/νε = 100, where the turbulent velocity fluctuations reach their maximum level before dropping to zero at the wall. Accurate

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modelling of the increase in velocity fluctuations with the distance from the wall remains a challenge. One way to bypass the frustrations associated with the accuracy of the turbulence models is to go directly towards LES or DNS to resolve the behaviour of such small fluctuations at the sub-grid scale level. Towards this end, IAR has developed LES and DNS methodologies to investigate such viscous flows for typical channel flow configurations. Flows past simpler Cartesian-based configurations with straight walls have been adopted to avoid the fine grid requirements associated with more complex aircraft wing body configurations, although some computations on flows in cylindrical geometries have been attempted.

While considering incompressible flows at low Reynolds numbers for medium- to micro-sized air vehicles, one also has to contend with flows past moving surfaces. Consequently, one is confronted with the challenge of synchronizing the moving portion of the computational mesh to accurately reflect the unsteady fluid surface interaction when the flow field solution is marched in time. At IAR, we have demonstrated such capability for simple airfoils, executing a variety of pitching and flapping motions. This demonstration is still a far cry from the more intricate wing motions associated with insect or bird flights. The wings of a housefly, for example, execute a complex sweeping motion schedule. The insect wing roots hinge to themselves and translate to and fro on the body to provide an effective local angle of attack to an individual wing at a given position. Each wing, in a typical computational domain, must slide smoothly inside a parent mesh, which must itself be moving at the speed of the housefly. Even without a facility to simulate the elastic deformation of the wing, the rigid body simulation requires a huge grid generation effort and advanced grid-movement technique for a typical Navier-Stokes computation.

Low Reynolds number flows are strongly influenced by the nature of transition from laminar to turbulent conditions. On a simple airfoil, if the transition is not predicted accurately, an inadvertent simulation of a thicker turbulent boundary layer near the leading edge may lead to erroneous predictions of the separation bubble, causing inaccurate lift and drag results. Towards this end, IAR is developing a numerical technique for solving the Orr-Sommerfeld equations to study stability and transition phenomena in different laminar fluid flow problems. Such problems include Balsius and plane Poiseuille flows. The stability analysis is carried out by

imposing infinitesimal perturbations to the basic laminar flow. This technique is used to determine the thresholds above which the steady laminar flows become unstable to small perturbations. An algorithm for solving the Orr-Sommerfeld equations in the form of an eigenvalue problem was obtained on the basis of the finite element method. The numerical results, obtained with uniform and non-uniform meshes using cubic or quintic Hermite elements, showed excellent agreement with the results available in the literature for plane Poiseuille flow.

2 Unsteady Flows past Oscillating and Stationary Wings

2.1 Description of Numerical Algorithms

IAR/NRC has developed an in-house code for computing unsteady incompressible flows. The integral form of the conservation law for mass and momentum was used to describe a flow in moving coordinates. For an arbitrarily shaped domain of moving control volume V bounded by a closed surface A, the continuity and momentum equations are written as follows:

In the equation, represents the fluid density, the fluid velocity, the velocity of grid movement, and p the pressure. Also, is the unit vector orthogonal to the surface A and is the unit tensor. The stress

tensor can be written as ,

where is the fluid dynamic viscosity and the rate of strain (deformation) is defined as

A fully implicit temporal differencing scheme was used in the discretisation, which made the algorithm stable for large time-steps. The discretisation of the convective and diffusive fluxes was carried out in a co-located variable arrangement using a finite

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volume approach. The coupling of the pressure and the velocity was handled using the SIMPLE algorithm [3]. The continuity equation was transformed into a pressure-correction equation, which had the same general form as the discretized momentum equations. The use of the co-located variable arrangement on non-orthogonal grids required that the SIMPLE algorithm be modified slightly to dampen numerical oscillations. A pressure-velocity coupling method for complex geometries used by Ferziger and Perić [4] was implemented, where an additional pressure gradient term was subtracted from the velocity value at the surface of the control volume. The space conservation law

was imposed in the pressure-correction equation for moving grids to ensure the conservation of mass. More details of the present procedure can be found in [5].

To take into account the turbulence effects, the RNG variant of the k- turbulence model proposed by Yakhot et al. [6] was adopted. To enable LES practices for complex geometry flows, the Smagorinsky model was also implemented in the in-house code.

2.2 Computation of Flows over Oscillating Wings

Numerous calculations focused on the flows past stationary and oscillating wings were performed. For simplicity, two test cases of oscillating wings are illustrated here. The first wing was rectangular in plan form with a 1-foot chord and 2-foot span, based on the NACA 0012 airfoil profile. Halfman [7] has reported experimental results for this wing (airfoil) in pitching motion. The wing was subjected to a freestream velocity of = 36 m/s, and was dynamically pitched about an axis, located at the 37 percent chord, with a reduced frequency of

, where b=c/2 is the semi-chord and is the angular frequency of the pitching motion. The corresponding Mach and Reynolds numbers were 0.11 and , respectively. The amplitude of the motion was set equal to 6.74o

and centred on . In addition to the calculation of the pure pitching motion, another case with an in-

phase flapping oscillation was performed. In the calculations, symmetry boundary conditions were used in the spanwise direction.

As seen in Fig. 1, an O-H-mesh was used in the calculations. The mesh size was 1214041 grid points for the chordwise, normal to the wall and spanwise directions, respectively. The outer boundary was ten chords away from the wing. The inner part of the O-mesh was oscillated periodically together with the wing.

Fig. 1 Computational mesh of a NACA 0012 wing configuration in an oscillating motion.

Figure 2 shows that the integrated lift coefficient at the mid-span in the case of combined pitching/flapping motion differs significantly from the values for pure pitching motion. This indicates that the flapping motion changes the dynamic performance of an aircraft wing and is an important technical perspective of an application to a real micro UAV flight with high lift.

Fig. 2 Lift coefficient showing strong deviation between pure pitching motion and combined pitching/flapping motion.

To understand the deviation between the integrated results of pure pitching and pitching/flapping motions, the pressure distribution at zero angle of attack during the pitching up period is depicted in

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Fig. 3. As shown in the figure, the flow field in the case of combined pitching/flapping motion changed more violently than in the case of pure pitching motion. The flapping motion produced stronger pressure suction at the upper wing surface during the flapping down period.

Fig. 3 Pressure distribution on the airfoil surface at =0, during pitching up period. Top: pure

pitching motion; bottom: combined pitching/flapping motion, flapping down period.

To demonstrate the capability of predicting the flow field details around the wing tip, calculations were performed for another wing in a pitching/flapping motion using the same parameters and the RNG k- turbulence model. However, the wing was assumed to have a one-foot span and its root was mounted on a flat plate that was oscillating together with the wing. The computational domain was extended from the wing tip to the far field by three spans. As expected, the tip vortex induced by the movement of the wing flapping was very strong at the maximum altitude of the flapping motion, see Fig. 4.

2.3 Computation of Flows past a Stationary SD7003 Airfoil (Wing) at very Low Reynolds Numbers

IAR/NRC carried out a parametric study of quasi-2D (Q2D) LES (only 4 grid points used in the span-wise direction) computations on low Reynolds num-ber flows past an SD7003 airfoil at Re = 60000. Three different meshes (M1, M2, M3) and three dif-ferent sub-grid scale (SGS) concepts were con-sidered, namely, the traditional Smagorinsky SGS

model, the selective mixed scale model (SMSM) Model [8], and an implicit SGS scheme. The impli-cit approach used an upwind scheme and did not ex-plicitly use an SGS model. The mesh M2 had the same number of grid points as M1 but with a denser points concentration on the suction side.

Fig. 4 Flow field around a NACA 0012 wing tip at different instants of time during one pitching/flapping cycle;

, .

The calculations confirmed that the grid resolution on the suction surface played a significant role. The Q2D LES calculations captured the leading edge laminar separation, followed by a transition and

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turbulent reattachment on the mesh M2, see Fig. 5. The Q2D LES calculations confirmed that for the same Reynolds number, the laminar separation occurred earlier at higher angles of attack than at lower angles of attack, with a decrease of the laminar separation bubble length as the angle of attack increased. Unfortunately, the predicted laminar separation bubble was about 50% larger than the one observed in the experiment. It is believed that the turbulent momentum transport is strongly affected by eddies with rotation not aligned with the spanwise coordinate, which should be numerically resolved. Vortex shedding from the primary separation bubble was also observed, as demonstrated in Fig. 6.

Fig. 5 Velocity vectors over the SD7003 airfoil at Re = 60000 and = 8. Time-averaged quasi-2D LES results using the SMSM.

Fig. 6 Instantaneous velocity vectors and spanwise vorticity around the SD7003 airfoil at Re = 60000 and = 8.

3 LES of Turbulent Flow in a Square Annular Duct and a Confined Square Coaxial Jet

3.1 Computational Strategy

Turbulent flows can be investigated with any of the following approaches:

(1) Direct numerical simulation (DNS);(2) Large eddy simulation (LES);(3) Reynolds-averaged Navier-Stokes method

(RANS).

In the present work, the spatially filtered, three-dimensional, time-dependent Navier-Stokes (NS) equations are used, which are closed using an SGS model. These equations in non-dimensional form are:

where the indices i, j, k = 1, 2, 3 refer to the x, y and z directions, respectively; x is the streamwise direction and y and z are the transverse directions. Equations (4) and (5) are a simplified form of Eqs. (1) and (2) in a stationary coordinate system.

The governing equations were spatially discretized using a second-order finite volume method on a staggered grid. As demonstrated by Kim and Moin [9], a common practice for the temporal discretization of the LES governing equations is to apply the second-order Adams-Bashforth scheme for the convection terms and the second-order Adams-Moulton scheme for the diffusion terms (Stoer and Bulirsch [10]). The fractional step method of Kim and Moin [9] was used to decouple the pressure and velocity and to obtain the time-dependent pressure and the divergence-free velocity. Thus, the discretized governing equations can be expressed as:

,

where:

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3.2 LES of Turbulent Flow in a Square Annular Duct

The LES method was applied to a square annular duct flow in Xu et al. [17]. This is a complex flow, compared with a square duct, because the square annular duct contains both concave 90 corners around the outer square duct and convex 90 corners near the inner square duct, see Fig. 7. The turbulence structures, particularly the turbulence-driven secondary flow structures, are more complex than those in a square duct. In the current investigation, details of the turbulence-driven secondary flow in a square annular duct were explored and found to contain a chain of counter-rotating vortex pairs symmetrically placed around the bisectors of both concave and convex 90 corners, as shown in Fig. 8. The turbulence statistics were performed, which made it possible to directly interrogate the origins of the turbulence-driven secondary flow. A relationship between the mean streamwise velocity and the distance from a concave 90 corner along the corner bisector, as shown in Fig. 9, was derived by curve-fitting the results from both an LES in a square annular duct and a DNS in a square duct. We argue that this relationship is universally valid for flows near a concave 90 corner as it matches the DNS and LES data very well.

Fig. 7 Flow in a square annular duct.

Fig. 8 Turbulence-driven secondary flow in a square annular duct.

Fig. 9 Universal logarithmic streamwise velocity profile near concave and convex corners.

3.3 LES of Turbulent Flow in a Confined Square Coaxial Jet

An LES simulation was performed to investigate the spatial evolution of the turbulent flow in a confined square coaxial jet as shown in Fig. 10, see Xu et al. [18]. The turbulent inflow conditions in the jet were obtained from LES in both square and annular ducts using a temporal approach. Such prescription of the inflow boundary conditions faithfully represents the turbulent inlet conditions and makes it possible to realistically investigate two types of turbulent mixing mechanisms originating from the streamwise shear, caused by the streamwise velocity difference, and the secondary shear, induced by the turbulence-driven secondary flow, see Fig. 11. The turbulent mixing properties in the confined square coaxial jet

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were studied by analyzing the spatial evolution of the mean flow field and the second-order turbulence statistics. The simulation results presented reasonable agreement with experimental data from a square free jet and the measurements of a confined plane jet, as shown in Fig. 12. The turbulent mixing phenomena were interrogated using the streamwise vorticity distributions on various section planes of the instantaneous flow field. The principle of dynamics in Zaman [11] was used to understand the effects of the turbulence-driven secondary flow and to explain the observation found in the simulation.

Fig. 10 Flow in a confined square coaxial jet.

Fig. 11 Turbulence-driven secondary flow in the in-let of the confined square coaxial jet.

Fig. 12 Streamwise velocity profile along centerline of the confined square coaxial jet.4. Stability Analysis of Laminar Flows

4.1 Finite Element Solution of the Orr-Sommerfeld Equation

For plane Poiseuille flow and Blasius boundary layer stability analysis, the velocity perturbation can be introduced as:

and the Orr-Sommerfeld equation is:

where D=d/dy and is the streamwise wave number. The parameter is a complex number defined as , where is the circular frequency and is the amplification factor. The instabilities set in when and are damped when

. The dimensionless form of the Orr-Sommerfeld equation is obtained using a length scale b=h (half height of the channel for Poiseuille flow) or (displacement thickness for the Blasius boundary layer). The Reynolds number is defined as

, where U is the maximum velocity of the main flow.

The boundary conditions are

The solution of the Orr-Sommerfeld equations was obtained numerically using a finite element method. As the Orr-Sommerfeld equation is a fourth-order differential equation, high precision Hermite finite elements were used. Quintic elements were considered in the present study. Using the Bubnov-Galerkin method, the discretized form of the Orr-Sommerfeld equation led to the following eigenvalue problem:

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The IMSL library was used to compute all the eigenvalues for a given number of finite elements. The present results were compared with the most accurate results reported in the literature by Orszag [12] for plane Poiseuille flow, and an excellent agreement was obtained [13]. The stability diagram for plane Poiseuille flow is displayed in Fig. 13, and the stability diagram for the Blasius boundary layer is illustrated in Fig. 14. The results were obtained using a temporal theory. The spatial linear theory coupled with the well-known en method is presently being validated for the Blasius boundary layer to predict the onset of transition to fully turbulent flows.

0.0 1.0 2.0 3.0 4.0 5.00.4

0.6

0.8

1.0

1.2

Re x 10-4

i

Fig. 13 Stability diagram for the plane Poiseuille flow.

Re

500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

i

Fig. 14 Stability diagram for the Blasius boundary layer flow.

5. Conclusions

Even without strong fluid/structure interactions, viscous separated flows are very difficult to model at low Reynolds numbers owing to the unpredictable

nature of the variety of vortical structures present in the flow. In this paper, viscous unsteady flows passing stationary and moving airfoils and wings have been attempted using two equation turbulence models and large eddy simulation. Moving surfaces give rise to separated vortical flow structures, which cannot easily be simulated and may be responsible for differences between predicted and measured results.

Large eddy simulation techniques have been applied to demonstrate the superiority of this approach to conventional two-equation models. Results obtained in complex concave and convex corners of the annular duct and the confined square co-axial jet flows have compared well with measurements and other results in literature.

A Hermite finite element approach has been developed to predict transition to turbulent flow in two-dimensional Poiseuille and Blasius flows. Results obtained to date provide encouragement for us to extend the application of this method to more complex curvilinear geometries using the en method.

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