Minutes of the Meeting on 1 May 2001

Present:     Keith Austin (Flowmaster), Arris Tijsseling (Eindhoven University), Bruno Brunone (Perugia University), Mohamed Ghidaoui (Hong Kong), Zhao Ming (Hong Kong), Della Leslie, Alan Vardy, Jim Brown and (Dundee University).

Chairman:                   Alan Vardy

Minutes:                   Della Leslie

Meeting commenced at 09:30 hours.

Chairman’s Introduction:

Welcome to all. Alan summarised the connection between the two theories on unsteady friction. Today should give an overview of the methods and show that progress has been made since the last meeting. Alan distributed a revised agenda.

Item 1.                   Improved unsteady friction formulae, part I

Jim Brown presented some of his and Alan’s current work on improving unsteady friction formulae. He began by giving a brief history: aim to better understand problem and to convert to rough from smooth pipe. Work on rough pipe not covered today, but are very close to a completed solution.

General interpretation of unsteady friction:

Fast transients in flow occur, e.g. valve closure or in tunnels during the passage of pressure waves. To predict friction requires the numerical analysis of equation, but in real systems this is not possible because of prohibitive computation cost (e.g. 60hr/case), therefore assume axi-symmetric (1-D) to reduce cost.

1-D unsteady flow: Incompressibility is assumed and Jim presented a general form of the equations of motion. This general form covers both plane and axi-symmetric coordinates. It is assumed turbulence properties don’t have time to change and viscosity is a function of position. A Laplace transform is used on the equations and initial velocity is assumed to be zero. The result provides the transformed wall shear stress, from which the steady-state condition is found by letting the parameter s (from Laplace) tend to zero. Subtracting this from the original expression provides the unsteady component of the (transformed) shear stress. To find the actual shear stress we need to perform an inverse Laplace transform, and Jim said that this is not always possible to calculate. Next Jim describes a new development of a pressure gradient based weighting function, whereas usually acceleration has been used. He explains how this weighting function can be derived from the preceding analysis – providing a transformed weighting function. Jim shows how this new weighting function can be related to the conventional weighting function, illustrating both have a very similar form. Then, instead of trying to directly invert the functions, an approximation for each weighting function is defined with known inverses. He illustrates that the approximation is very accurate when compare to the actual function, and provides a relatively simple final formulation. He noted that the weighting functions rapidly decrease with time. By defining the time for it to reach 95% of its final value, if the time step is larger than this then the weighting function has very little effect on the solution. Jim illustrates the method uses the Zielke case – it clearly demonstrates the effectiveness of the method.

Item 2.       2-D Turbulence Models: Use steady-state based

Mohamed Ghidaoui first gave the group three current/new papers he had written (Della will photocopy and distribute to the group with the minutes). His presentation took the form of two parts, the second part later in the day – item 4. The first part examined the use of steady state based 2-D turbulence models. It is assumed the general problem: non-slip condition at wall, fast transient, short time scale. He then examined a number of different steady-state based turbulence models for waterhammer, to test for sensitivity of results to eddy viscosity distribution. He illustrated the various viscosity profiles the models used and compare results. It showed there was very little sensitivity to data near the centre of the pipe; saying was probably because dominant behaviour is near pipe wall. Mohamed concluded that there is a lack of sensitivity and thus justifies the simplified distribution of Vardy and Brown 1995 and explains why Vardy and Brown model as accurate as 2-D models (high numerical resolution of centre not is needed).

Mohamed continued, saying that behaviour was dependent on time-scales: time scale of transient and time scale of diffusion. If transient scale is greater than diffusion, then turbulence frozen (everything localised near the wall). Mohamed proposes a dimensionless parameter, P, to decide when turbulence is frozen, i.e. time scales different enough. He than demonstrates the use of the parameter by giving a figure illustrating the error between experiments and various models. This showed the best models are associated with the highest P.

Bruno promised to send actual experimental data (Mohamed had been trying to extract information from figures in published work).

Item 3.                   Large Initial Reynolds number Transients

Bruno Brunone gave an interesting presentation on some of his resent research development. First he showed some figures, based on 2-D numerical model, giving values of the parameter k vs. Reynolds number, for each y₀ (Alan and Jim pointed out that the definition of y₀ could be expressed as the friction(2fL/D)*Mach number). Figures had a similar form to the Moody diagram. He said that available experimental data used only limited Re values, so was unable to compare.

Next he described the current problem he had been working on – he had been approached by a water supply Company who had problems with a rising main. They had problems with pipe movement (pipe passes through basement of a building).

Bruno had obtained experimental field data associated with the problem. Transient tests due to pump shutdown had Reynolds number of: 240568, 269476 and 390005 (three tests), much higher than in many past paper (e.g. Budny has about 11000). The experimental results gave data similar to lab test, showing decay and rounding of peaks. In the third experimental test, there was water-column separation, which Bruno said was due to a new pump with low inertia.

The next stage was the numerical modelling of the field tests. Bruno presented the approach he used to model the system – based on the unsteady friction model using J_u = k₃/g(dv/dt-a*dV/ds). He gave the resulting MOC equations, including the parameter k₃ (appears in only one equation). The equations are different along the each characteristic, C+, C-. Alan said that direction of flow is important with respect to axes. Alan suggested, you need to refer to positive and negative velocity, i.e. choose axis in direction of positive velocity and therefore can stay with non-symmetric equations. Bruno then presented a symmetric version of the equation, which includes sgn(V) term - this forces k term to appear alternatively in the each characteristic. The parameter k₃ is unknown for large Reynolds numbers, and the values from Vardy and Brown smooth pipe data are assumed. Despite the problem having rough pipes and larger Re, the results found appeared reasonable – showed decay and rounding. A discussion on the differences developed. Alan pointed out which regions of the plots were important with respect to unsteady friction, showing there were in fact significant differences.

Item 4        2-D Turbulence Models: Assume Symmetric flow

Mohamed Ghidaoui presented the second part of his presentation on 2-D turbulence models. This was examining the assumption of symmetric flow. First he showed experimental results from a paper: this described a ramp type experiment (acceleration, maintain and decelerate the fluid). Results showed the unsteady velocity profiles. The first picture showed oscillation near walls (laminar flow). The second included vortices in the flow– these were not symmetric and showed a phase lag (helical type). It showed these vortices growing and interacting, illustrating a change of flow from laminar to turbulent; the instability started near inflexion points. The last example again illustrated the vortices, still with a phase lag, but this time there was no interaction. Mohamed refer to experimental results by Bruno, giving non-symmetric velocity profiles, and said that this was consistent with helical form. The axi-symmetric modes are very unstable.

Next Mohamed described stability mechanisms in the system: If there is more input than output then there is a need for the flow to change, i.e. stability governed by production and dissipation of energy. He showed that production of energy is greatest at the inflexion point and depends on the strength of the transient. He presented some stability diagrams and showed that experimental results agreed with the diagram (only limited data available) and most work did in fact fall in the stable region (hence symmetric). Mohamed concluded the presentation, saying that experimental evidence showed flow instability; and the existence of flow instability & impact on energy dissipation has not been recognised in the past.

Item 5                   Improved unsteady friction formulae, part II

Jim Brown gave the second part of his presentation on an improved unsteady friction formula. He outlined some of the most recent work, which uses an annulus formulation for viscosity. In the outer region (annulus), Cartesian (plane flow) coordinate system in used, whilst the inner core uses polar coordinates (axi-symmetric). The equations presented in the first part of the presentation (Item 1) allows for both systems. In the annulus, viscosity is assumed to be linear (with position) and in the inner core is assumed constant - this can be any value be will examine the case where it matches the outer value of annulus. The boundary conditions assume non-slip conditions at the wall, and velocity is continuous between annulus and the core. The velocity is prescribed at the interface and there in equality between velocity gradients of the core and annulus at the interface. In the 1995 paper, a uniform velocity was assumed in the core region and this is the main change in the current work.

Using the method described in item 1, Jim shows that the solution for the problem may be found – giving a pressure gradient based weighting function. He compared the new solution with that from the 1995 paper. Using a log scale there appeared to be little difference, but with a linear scale there was a clear change. The current work increased the time-life of unsteady friction. It introduces changes in the later stages of models, which was where there previously had been problems, and introduced more wall shear stress.

Item 6                   Turbulence model in quasi-2D simulations of pipe flows

Zhao Ming gave an interesting presentation that compared turbulence models in quasi-steady 2D simulations of pipe flows. First he outlined the difference between quasi-steady and frozen turbulence model. Quasi-steady: instantaneous turbulence for unsteady flow assumed same as steady flow, steady turbulence and slow transient with very short time. Frozen turbulence: turbulence assumed frozen and initial steady eddy viscosity. There was a discussion on the different approaches. Alan said that in his work, computer simulation use local Reynolds number, so that it was similar to quasi steady but freezing each time step.

Zhao proposed a new approach for the quasi-steady model, which he called the decomposition model. In this he divided the velocity into two parts: initial steady velocity + ‘unsteady’ part. Alan said that it would be better if the ‘unsteady’ was given an alternative name, because it is not used in the same context as other work. By removing the initial steady profile from the analysis the remaining component tends towards a log profile (with time) and has no inflexion points – this fitted better with assumptions made for the quasi-steady model. Zhao showed comparisons between the models and with experimental. The decomposition model matched the peaks (produced more damping than other models) in experimental data very well.

Alan asked what are the benefits of the model, and whether there could be problem in the limits (subtracting two large numbers – both steady and final ‘unsteady’ components are non-zero)? A general discussion developed on the model.

Other business: Next meeting? It was proposed that the next meeting should occur in about a year’s time – should contact participants near the time.

Item 7.                   Chairman’s Closure

Thanks to all participants

Closure of meeting at 16:50 hours.

Quote of the day: Engineers observe what they can’t solve; Mathematicians solve what they can’t observe.