Turbulence

Many real flows are turbulent in nature e.g. flows around an aircraft wing, boundary layers on wings of aircraft, jet streams and combustion. Turbulence is a difficult phenomenon to define but generally has the following characteristics such as being associated with large Reynolds number, irregular in nature, diffusive, possessing three-dimensional vorticity fluctuations and being dissipative. The rate of mass and momentum transfer in turbulent flow is much higher than that in Laminar flow.

Launder, (1991) gives the definition of turbulent flow as “at moderate Reynolds numbers the restraining effects of viscosity are too weak to prevent small, random disturbances in a shear flow from amplifying. The disturbances grow, become non-linear and interact with neighbouring disturbances. This mutual interaction leads to a tangling of vorticity filaments. Eventually the flow reaches chaotic non-repeating form describable only statistical terms. This flow is turbulent flow”.

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Since Osborne Reynolds, it has well been known that flow behaviour can be categorised by the critical Reynolds number which governs the onset of turbulent flow. Based on the experimental work that was done by Reynolds on pipe flow and flow on a flat plate, the values for the various regimes were obtained. Thus, the flows can be effectively sub-divided into Laminar or turbulent flow.

The Navier Stokes equations given in Equations (2) are applicable to all fluid flow and thus would be able to handle turbulent flows in theory. Unfortunately, the actual numerical solutions to turbulent flows are more complex than for laminar flows because for turbulent flows the discretization of small steps in the flow in both time and space is important, which must to be performed down to the Kolmogorov scale, given by:

ᶯ = (𝑣 3 𝜖) 1 4 (9)

Nevertheless, it is accepted that the Navier Stokes equations can be used to directly solve turbulent flow fluid problems. This approach of modelling is generally called Direct Numerical Simulation. This fluid modelling approach to turbulent energy cascade by using the Navier stokes equations requires that the turbulent length scales be larger than the grid spacing. Direct Numerical simulation tries to simulate all the scales of the turbulence itself. The grid size, quality and the maximum permitted time step for a Direct Numerical Simulation must be small enough to capture the Kolmogorov scales of the turbulent flow (Nichols, 2009). Direct Numerical Simulation is characteristically unsteady thus, it requires a long period to run to ensure that the obtained result for the simulation is statistically stable and independent of the initial conditions inputted at the beginning of the simulation. Direct Numerical Simulation requires grids that are uniformly spaced because otherwise the higher order numerical algorithms in Direct Numerical Simulation are invalid. Grid stretching should also be avoided as it reduces the order of the numerical grids and increases the numerical dissipation of the algorithm. It requires extremely large computational grids even for small Reynolds number simulation which could be a problem. Direct Numerical Simulation has only been used for problems with low Reynolds numbers and on very simple geometries due to computer limitations in terms of processing speed and memory required to perform numerical simulation for higher Reynolds number problems. Due to the complexity of most engineering problems it is impossible to simulate such problems with the Direct Numerical method and therefore, some form of modelling of the turbulence is often required so that solutions to engineering problems can be obtained for large scale problems.

As an alternative, Large Eddy Simulation tries to modify the approach of the Direct Numerical simulation by only modelling the smallest turbulence scales (Kolmonogorove scales) in a computational fluid dynamic simulation

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(Breuer, 1998). These small turbulence scales could be modelled with simple turbulence models because they are nearly isotropic. In Large Eddy Simulations, large scales are computed explicitly. An advantage of this approach is that large eddies, which are hard to model in a general way, due to the fact that, these large scales are anisotropic are calculated directly (resolved directly) while the small eddies are modelled since they are closer to isotropic and can easily adapt to maintain a dynamic balance with the rate of energy transfer imposed on the flow by the large eddies. In Large Eddy Simulations, the turbulence scales are resolved down to the inertial sub-range (Andersson et al., 2011). As a general rule of thumb suggested by Andersson et al., (2011) at least 80% of the turbulence energy should be resolved in the calculated velocities. Large Eddy Simulation are unsteady, they require to be run for large number of times steps in order to eliminate initial conditions dependency for the simulation to be statistically stable. Large Eddy Simulations can be used at much higher Reynolds number than Direct Numerical simulation. The governing equations for Large Eddy Simulations are obtained by spatially filtering over small scales. i.e. filtering the Navier Stokes equations of small scales. examples of the filters used are Gaussian and sharp spectral filters (Saguat, 2001). The filter function determines the division of the turbulent spectrum into the grid realised and sub-grid regions i.e. (the turbulent scales that needs to be calculated directly and the ones that needs to be modelled). It also requires a low dissipation numerical scheme.

The filtered continuity equation is given by 𝜕𝑈𝑗

̅̅̅̅̅ 𝜕𝑥𝑗

= 0

(10) And the corresponding momentum equation is given by

𝜕𝑈̅_𝑖 𝜕𝑡 + 𝜕𝑈̅𝑖𝑈̅𝑗 𝜕𝑥𝑗 = −1 𝜌 𝜕𝑃̅ 𝜕𝑥𝑗 + 𝑣 𝜕 2_𝑈 𝑖 ̅ 𝜕𝑥𝑗𝜕𝑥𝑗 − 𝜕𝜏𝑖𝑗 𝜕𝑥𝑗 . (11)

In Large Eddy Simulation closure problems arises due to the presence of the residual stress tensor 𝜏𝑖𝑗 also known

as the sub-grid stress tensor. This residual stress tensors describes the transfer of momentum by turbulence in the flow at turbulent scales that are smaller than the filter, and is given by:

𝜏𝑖𝑗= 𝑈̅̅̅̅̅̅ − 𝑈𝑖𝑈𝑗 ̅ 𝑈𝑖̅ 𝑗

(12)

The filtered velocities (𝑈̅ 𝑎𝑛𝑑 𝑈𝑖 ̅̅̅̅) are solved for in the equation but the correlation term 𝑈𝑗 ̅̅̅̅̅̅ is an unknown 𝑖𝑈𝑗

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involves modelling of a sub-grid viscosity (Andersson et al., 2011). An example of such models is the Smagorinsky-Lilly model (Canuto, 1994). In general, some challenges faced by using the Large Eddy Simulation is that it is still not capable of dealing with flight Reynolds number at reasonable turn-around times.

Hybrid RANS/LES models are a new kind of turbulence models which can be used for unsteady high Reynolds number flows. These models are extensions of the Large Eddy Simulation models (Nichols, 2009). The concept behind these models is to use a modified Reynolds Average Navier–Stokes turbulence model as a sub-grid model (i.e. turbulent scales that cannot be achieved on a computational grid) for the Navier-Stokes equations. The main aim of this model is to use Reynolds Average Navier-Stokes Simulations to calculate the boundary layer where very small turbulent scales are present, then use a Large Eddy Simulation type model to simulate the smaller turbulence scales that are away from the body used for the simulation; therefore, the large turbulent scales that are located away from the body are simulated using the unsteady Navier-Stokes equations. This model serves as a means to close the gap between the Reynolds Average Navier-Stokes Simulation model and the Large Eddy Simulation model. These models require a coarser mesh than the Large Eddy Simulation because the Reynolds Average Navier-Stokes Simulation type sub-grid model is valid for non-isotropic turbulent scales (Nichols, 2009). This model requires “spatial filtering of the Navier Stokes equations in order to determine the local values for the sub-grid turbulent viscosity” (Nichols, 2009). It does not require very low dissipation thus, so it can be easy to implement.

The following section of the report presents an overview of developments of the Reynolds Averaged Navier Stokes form of the Navier Stokes equation. This is discussed here as they are used in the numerical simulations presented in this report.

In document An Experimental and Theoretical Investigation of Novel Aircraft Drag Reduction (Page 93-96)