CFD simulations
Computational studies are carried out using DNS and RANS approaches with computer codes developed and maintained by the group.
DNS simulations
An in-house DNS code [1,2] has been developed and is being using for the simulations. Various unsteady flows (periodic / non-periodic) can be studied by the code.
A standard second order finite difference method is used to discretise the spatial derivatives of the governing equations on a rectangular grid, where a three-dimensional staggered mesh is employed with a non-uniform spacing in the direction normal to the wall. The non-solenoidal intermediate velocity field is evaluated by means of a low storage third-order Runge–Kutta scheme for the non-linear terms together with a second order Crank–Nicholson scheme for the viscous terms.
The time advancement of the Navier–Stokes equation is based on a fractional-step method described by Kim & Moin [3] and Orlandi [4] to enforce the solenoidal condition. The resulting discrete Poisson equation for the pressure is solved by an efficient 2-D FFT, taking advantage of an imposed periodicity in the streamwise and spanwise directions [4]. The Message-Passing Interface (MPI) is used to parallelize the code for use on our distributed-memory computer clusters.
The code is adopted for simulations of 2-D roughness using an immersed boundary method (IBM) [5] (the corresponding subroutines are kindly provided by Professor Paolo Orlandi, University of Rome, Italy). The code is currently being revised to treat a 3-D pyramid roughness.
Test cases
i) smooth-wall, mild acceleration
ii) smooth-wall, fast acceleration
- References
-
1. M. Seddighi, Study of Turbulence and Wall Shear Stress in Unsteady Flow Over Smooth and Rough Wall Surfaces, PhD thesis, 2011.
2. M. Seddighi, S. He, P. Orlandi & A. Vardy, A comparative study of turbulence in a ramp-up and a ramp-down flow, Flow Turbulence and combustion, 86 (3-4), pp. 439-454, 2011.
3. Kim J, Moin P. Application of a fractional-step method to incompressible Navier-Stokes equations. Journal of Computational Physics 1985; 59(2):308-323.
4. Orlandi P. Fluid Flow Phenomena: A Numerical Toolkit. Kluwer, 2001.
5. Fadlun, E.A., Verzicco, R., Orlandi, P., & Mohd-Yusof, J. 2000. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161, (1) 35-60.
RANS simulations
An in-house CFD code is used which was developed over several years for the specific study of unsteady turbulent flow in pipes. It is based on a finite volume/finite difference scheme with an explicit time marching algorithm.
The present results were generated using the low Reynolds number Launder-Sharma Eddy Viscosity Model which has been verified through many years of experimental research.
1-D simulations
Unsteady flows in pipes
The simulation of transient flows in pipes is a mature subject. High quality 1D software is widely available for applications ranging from domestic pipework to hydro-electric power. In most cases, flows along single pipes can be analysed with good accuracy and so the accuracy of a pipe network analysis is controlled by the representation of boundary conditions such as pumps, turbines, valves, surge tanks and specialized fittings.
In a few instances, however, the ability to model flows within individual pipes is a limiting factor. An obvious example is when the fluid cavitates and therefore ceases to be a simple liquid. Another example - of special interest here - is when attention focuses on the evolving shapes of individual wavefronts, not simply on their amplitudes.
In an overwhelming number of analyses, it is acceptable to express the underlying equations in a form that assumes that all disturbances travel at the same speed regardless of their amplitude. For example, in the most common method of analysis, namely the method of characteristics (MOC), it is usual to assume that the compatibility equations are applicable in the directions dx/dt = ±c, where c is the speed of sound and x & t are the distance and time coordinates.
In reality, the disturbances travel along dx/dt = U±c, where U denotes the mean velocity of the fluid. Thus waves with different amplitudes actually travel with slightly different speeds. It will be extremely rare for this particular approximation to have an important influence on predicted conditions in liquid flows, but it is an approximation nonetheless and it has significant consequences for analyses of gas flows.
A more important approximation is the assumption that various empirical coefficients used in analyses are independent of the state of flow. The most important example is the coefficient of friction. If this is assumed exactly constant in any particular pipe, friction will not cause any change in predicted wave shapes.
Even if it is assumed to vary with the Reynolds number of the flow, any influence of the predicted wave shape is unlikely to be detected. When allowance is made for the dependence of the friction on the flow history, however - i.e. not only its instantaneous state - more substantial changes in shape are predicted. They are not great, but they need to be taken into account in specialist applications where wave shapes are important.
Models of unsteady friction
In line with promises made in the original funding bid, this project is focussed on increasing understanding of the underlying causes of unsteady friction and its characteristic behaviour. However, it is hoped that the results of the research will subsequently be used to enhance methods that the researchers have published for simulating the phenomenon in numerical analyses. It is therefore useful to give a brief indication of the main features of current methods of allowing for unsteady skin friction. These may be grouped into three broad categories.
1) Quasi-steady approximations
In the most widely used software packages for simulating transient flows in pipes, the wall shear stress τw at any particular location along a pipe is assumed to be the same as it would be in a steady flow with the same local velocity U. Such methods may be expressed generically as:
τw = f{U}.
In some simulations, the function is regarded as a constant whereas in others it is regarded as dependent upon the Reynolds number. This simple method of approximating skin friction is highly successful. In part, this is because many real flows include large regions of steady or slowly varying flow. At any particular location, strong accelerations are experienced for only short periods.
2) Instantaneous acceleration
When account needs to be taken of the influence of accelerations on wall shear stresses, use is often made of a model that is usually attributed to Brunone and his team in Perugia, Italy. In this model, the shear stress is assumed to depend upon the instantaneous acceleration as well as upon the instantaneous velocity. The method may be expressed generically as:
τw = f{U,∂U/∂t}.
Strictly, all modern implementations of this method allow for the convective acceleration as well as for the temporal acceleration. Empirical coefficients are needed in the assumed forms of the function, but the method has enjoyed considerable success in reconciling differences between measured and predicted pressure histories in pipes. This is especially so in laboratory studies where Reynolds numbers tend to be quite small and so the influence of skin friction is relatively large.
3) Flow history
The most comprehensive methods of allowing for unsteady skin friction take account of the flow history over a long period before the current calculation instant. Interest in this method was stimulated by the successful work of Zielke (1968) in relation to laminar flows. This method can be expressed as:
τw = f{U,(∂U/∂t)0, (∂U/∂t)-1, (∂U/∂t)-2, (∂U/∂t)-3, ….. }.
This method is more comprehensive than the above approximations, but it has the obvious disadvantages of (i) having multiple unknown coefficients and (ii) requiring successive accelerations at each calculation point to be retained in the computer memory. The key to Zielke’s success in the laminar flow case was to deduce universal values for the coefficients, thereby eliminating empiricism from the methodology. This was a huge step forward. Zielke did not overcome the second disadvantage, but Trikha (1975) subsequently showed how this can be done - at least partially. Trikha’s method must not be used directly because it is valid only for tiny time steps, but he showed the way forward.
Zielke’s method of determining the coefficients is valid only for laminar flow - because it is based on an assumption that the viscosity does not vary in space or time. Much later, Vardy& Hwang realised that a similar method could be applied to flows with spatially varying (but not time-varying) viscosity and Vardy & Brown have subsequently developed this approach for increasingly realistic representations of viscosity distributions in turbulent flows.
Other authors have followed a similar approach and yet more authors have addressed improvements in Trikha’s methodology for reducing the computational overload of the basic methodology. Because the viscosity distribution is assumed not to vary in time, the Vardy-Brown method is commonly described as a frozen-viscosity method. It has always been obvious that the frozen viscosity assumption could not be valid for indefinitely long periods, but there has been little evidence about the actual period of approximate validity (if any).
Experimental evidence
Two types of experiment have been used to assess methods of allowing for unsteady skin friction. The simpler of the two is indirect. In this approach, measured pressure histories in water-hammer tests are compared with predictions based on the various numerical approaches. It has been found that close agreement can be obtained using the instantaneous-acceleration approach provided that suitable values are used for empirical coefficients.
With suitable choices, the method can reproduce measured rates of overall damping accurately. In some cases, it can also predict the shapes of pressure histories well, but some caution must be exercised in the interpretation of this outcome because it can be strongly dependent on factors such as valve types and rates of closure.
The biggest drawback of the indirect experimental approach is that skin friction accounts for only a small proportion of the overall pressure changes and so it is difficult to deduce truly reliable information from pressure histories. In principle, it is more useful to measure the wall surface friction directly. Before the present research, however, the only direct measurements at the wall surface known to the researchers were for sustained accelerations over relatively large times and the measured behaviour differed strongly from predictions based on assumed frozen viscosity.
It was suspected that experiments focussed on earlier periods of the acceleration would be in better agreement, but there was no direct evidence to support this expectation.
Previous projects
Previous projects by the researchers have provided powerful evidence from experiments and detailed CFD analyses that the frozen viscosity assumption is realistic during the early period of response to a change in the mean velocity. The duration of the period of approximate validity has been confirmed to be approximately equal to ½R/uτ , where R is the radius of the pipe and uτ is the friction velocity.
In practical simulations of individual wavefronts, it will commonly be the case that this period greatly exceeds the period for which the dependence on the historical flow condition is strong. That is, the assumption of frozen viscosity will commonly be adequate over short times even though it is known to be incorrect over large times. In simulations over longer time periods, however – e.g. involving multiple wave reflections – the approximation will be wildly in error and so the use of the more complex model of wall shear stress will be pointless and, perhaps, counter-productive.
One valuable outcome of the on-going research is a clearer understanding of the time-limitation of current methods of implementing the prediction method based on time-histories of acceleration. Such methods seek to predict only the unsteady component of the wall shear, recognising that this will tend to zero as a flow’s memory of a velocity change fades.
To do this, nominally-independent analyses are undertaken for the unsteady flow and for an imaginary, quasi-steady flow based on the same frozen viscosity. It is now known that the second of these (i.e. the quasi-steady flow analysis) is more time-limiting than the first. In principle, this knowledge could be used to justify the development of even more complex formulations for predicting the wall shear, but the practical benefits of following such an approach in a comprehensive manner would be unlikely to outweigh the corresponding disadvantages.
Current work
Previous projects have focussed on smooth-wall flows whereas most turbulent flows in pipes are rough-wall - or at least partially rough. This is an important focus of current work. It should be noted, however, that the influence of unsteady friction is typically smaller in rough wall flows than in smooth wall flows. In part, this is because the steady flow component is greater. In part, it is because turbulence delay times are smaller – i.e. the durations of periods of pseudo-frozen viscosity are shorter.
One application in which the influence of unsteady friction has been found to be significant is the steepening of wavefronts propagating along railway tunnels. In this application, however, it is found that an even stronger dependence is observed on the track type. Sometimes, rails are mounted directly onto a continuous concrete surface (slab track). Alternatively, however, they might be supported on sleepers on a bed of coarse gravel (ballast track).
In the latter case, wavefront steepening is arrested far more strongly than is the case with slab track construction. The difference between the two forms arises because of a second physical phenomenon that has no direct relation to skin friction, but is proving to be amenable to similar mathematical techniques for prediction purposes. This realisation has motivated an expansion in the scope of the original research that could have considerable practical benefit.