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Computations of periodic turbulent boundary layers with moderate adverse pressure gradient
Abstract
In this computational study a two-equation eddy-viscosity model of turbulence is used to predict the development of boundary layers with periodic variation of the free-stream velocity and time-mean adverse pressure gradient. Transport equations are solved for q, the square root of the turbulence kinetic energy and ζ, its dissipation rate. Comparisons with reliable experimental data on such flows verified that the model can satisfactorily reproduce the time-average prime variables as well as their amplitudes. The predicted boundary-layer integral parameters were also found to be in good agreement with comparable data in the literature. Additional comparisons are made with the computed development of the same flow at constant pressure. The effects of the pressure gradient isolated and studied in this way were generally found to be similar to those of nonperiodic flow in similar conditions.
... The QZeta model is the two-equation Q − ζ low-Re turbulence model developed by Gibson and Dafa'Alla' [8]. The model uses the square root of the turbulent kinetic energy, q = √ k, and its rate of dissipation, ζ =ε/2q, whereε is the isotropic dissipation rateε = ε − 2ν(∂ √ k/∂x j ). ...
... The original model for the isotropic turbulence reads [8] Dq ...
Periodically-oscillated pipe flows in which the bulk velocity is varied about a non-zero mean level (Ub = Ub0[1 + γ cos ωt]) are computed using a low-Reynolds-number k– turbulence model. Comparison is made with data for periodic pipe flow and it is shown that model is capable of resolving the principal features of the highly non-universal turbulence profiles that occur under conditions of harmonic forcing. There follows an examination of the frequency response of the phase-averaged turbulent kinetic energy, k, which is analysed in terms of its first harmonic variation (k(r, ωt) = k 0 + k 1 cos(ωt + ψ); k 0, k 1,ψ = f(r)). In confirmation of the stress-transport model results of Cotton and Guy [J. Hydraul. Res. 42 (2004) 293], it is found that the modulation of the turbulent kinetic energy, k 1/γ k0 first responds in a quasi-steady manner and then, with increasing frequency, exhibits resonant behaviour, which is itself succeeded by a frozen response at higher frequencies. The resonant condition occurs when the time scale of large-scale turbulence is an order of magnitude less than the period of the imposed oscillation. The paper concludes with a discussion of the parallels that may be drawn between the present results and the experimental study of Mizushina et al. [J. Chem. Engng. Jpn. 6 (1973) 487] in which the effect of external pulsation on the turbulence “bursting” process was investigated
In periodic flows near walls transport effects may be considerably larger than in a steady turbulent boundary layer. The question
explored in this contribution is, therefore, whether providing separate transport equations for each of the Reynolds stresses
consequently leads to a better modelling of a periodic flow's evolution than an eddy viscosity scheme whose constitutive equation
is inherently linked to the generation and dissipation terms being in balance (local equilibrium). Our conclusion is that,
while a stress-transport scheme is indeed better equipped to reproduce the phenomena, it does not consistently out-perform
the EVM over the range of flows studied. In some cases it is suggested that more attention must be paid to modelling diffusive
transport in order to secure the benefits of second-moment closure. To illustrate sensitivity to diffusive transport, two
different diffusion models are tested, one of which leads to different effective transport coefficients in each stress component.
The generality of the energy-dissipation model of turbulent heat and
momentum transport, developed by Jones and Launder (1972, 1973), is
tested by applying the model to predict the swirling flows generated by
a rotating disk in a quiescent atmosphere. It is found that this
turbulence model can predict accurately the flow, heat transfer, and
mass transfer near a rotating disk. This is significant for the problem
of predicting convective heat transfer rates in turbine disks.
This study deals with turbulent oscillatory boundary-layer flows over both smooth and rough beds. The free-stream flow is a purely oscillating flow with sinusoidal velocity variation. Mean and turbulence properties were measured mainly in two directions, namely in the streamwise direction and in the direction perpendicular to the bed. Some measurements were made also in the transverse direction. The measurements were carried out up to Re = 6 × 106 over a mirror-shine smooth bed and over rough beds with various values of the parameter a/ks covering the range from approximately 400 to 3700, a being the amplitude of the oscillatory free-stream flow and ks the Nikuradse's equivalent sand roughness. For smooth-bed boundary-layer flows, the effect of Re is discussed in greater detail. It is demonstrated that the boundary-layer properties change markedly with Re. For rough-bed boundary-layer flows, the effect of the parameter a/ks is examined, at large values (O(103)) in combination with large Re.
The turbulent boundary layer driven by a freestream velocity that varies sinusoidally in time around a zero mean is considered. The flow has a rich behavior including strong pressure gradients, inflection points, and reversal. A theory for the velocity and stress profiles at high Reynolds number is formulated. Well-resolved direct Navier-Stokes simulations are conducted over a narrow range of Reynolds numbers, and the results are compared with the theoretical predictions. The flow is also computed over a wide range of Reynolds numbers using a new algebraic turbulence model; the results are compared with the direct simulations and the theory.
The results of Ramaprian and Tu (1982) on fully developed periodic
turbulent water flows in pipes are summarized and compared with those
obtained for time-mean zero-pressure-gradient periodic turbulent
boundary layers in the periodic-flow water tunnel at the University of
Iowa. The data are presented in graphs and characterized. At high
frequency and amplitude, small but detectable time-mean-velocity effects
with similar Stokes-layer thickness and near-wall phase angle about 10
deg are observed in both regimes.
In this paper, selected findings of a detailed experimental investigation are reported concerning the effects of forced free-stream unsteadiness on a turbulent boundary layer. The forced unsteadiness was sinusoidal and was superimposed locally on an otherwise-steady mainstream, beyond a turbulent boundary layer which had developed under constant-pressure conditions. Within the region over which free-stream unsteadiness was induced, the sinusoidal variation in pressure gradient was between extremes of zero and a positive value, with a positive average level. The local response of the boundary layer to these free-stream effects was studied through simultaneous measurements of the u - and v -components of the velocity field
Although extensive studies of unsteady, turbulent, fully-developed pipe and channel flow have been carried out, the problem of a developing turbulent boundary layer and its response to forced free-stream unsteadiness has received comparatively little attention. The present study is intended to redress this imbalance and, when contrasted with other studies of unsteady turbulent boundary layers, is unique in that: (i) it features an appreciable amplitude of mainstream modulation at a large number of frequencies of forced unsteadiness, (ii) its measurements are both detailed and of high spatial resolution, so that the near-wall behaviour of the flow can be discerned, and (iii) it allows local modulation of the mainstream beyond a turbulent boundary layer which has developed under the well-known conditions of steady, two-dimensional, constant-pressure flow
Results are reported which allow comparison of the behaviour of boundary layers under the same mean external conditions, but with different time dependence in their free-stream velocities. These time dependences correspond to: (i) steady flow, (ii) quasi-steadily varying flow, and (iii) unsteady flow at different frequencies of mainstream unsteadiness. Experimental results focus upon the time-averaged nature of the flow; they indicate that the mean structure of the turbulent boundary layer is sufficiently robust that the imposition of free-stream unsteadiness results only in minor differences relative to the mean character of the steady flow, even at frequencies for which the momentary condition of the flow departs substantially from its quasi-steady state. Mean levels of turbulence production are likewise unaffected by free-stream unsteadiness and temporal production of turbulence appears to result only from modulation of the motions which contribute to turbulence production as a time-averaged measure.
Measurements in turbulent channel flow with forced oscillations covering a wide range of frequencies (ω++/ = 2/ls+2). In the regime of high-frequency forcing (ls+ < 10) the oscillating flow u and τ are governed by purely viscous shear forces although the time-mean flow is fully turbulent. This behaviour may be explained by the physical significance of ls+. At lower frequency ls+>10, the oscillating flow is influenced by the turbulence, in particular the amplitude of τ increases with respect to the Stokes amplitude and becomes proportional to ls+. The relative amplitude of 〈u′u′〉 and 〈τ′τ′〉 decreases sharply with increasing forcing frequency once ls+<25. This decay of the turbulence response is faster for the wall shear stress. For forcing frequencies such that ls+<12, 〈u′u′〉 and 〈τ′τ′〉 lag behind 〈u〉 and 〈τ〉 by respectively about 75 and 130 viscous time units. These lags decrease by a factor 2 at higher forcing frequencies. It is shown that in the log layer, the turbulence modulation diffuses away from the wall with a diffusivity equal to that of the time-mean turbulence. The imposed oscillations are felt down to the small scales of the turbulence as may be evidenced from the cyclic modulation of the Taylor microscale, the skewness and the flatness factors of ∂u′/∂t. The modulations of the skewness and the flatness go through a maximum around ls+ = 12.
It is common knowledge that neither of the equations in the k-ε model is well-adapted to integration through the viscous layers to the wall. This paper shows how some of the difficulties may be alleviated by the choice of alternative variables.
Experiments show that pressure gradients have a strong effect on the law of the wall for temperature but have comparatively little effect on the law of the wall for velocity. This is contrary to the predictions of the mixing-length formulas, though these can be derived by apparently plausible dimensional analysis. To model this behavior, the choice of the transport equation for the turbulence-length scale, or its equivalent, becomes crucial; this is true both for two-equation models and for full second-moment transport models. This paper is a brief review of the experimental evidence, followed by a demonstration that the most popular 'length-scale' variable, the dissipation rate epsilon, and its temperature-field equivalent epsilon(sub T) fail to reproduce the observed behavior. A more general length-scale variable can be defined as k(sup m) epsilon(sup n), where k is the turbulent kinetic energy. It is shown that, overall, the omega variable (m = -1, n = 1) seems to be the best choice, with its temperature-field equivalent omega(sub T) for heat transfer, although there is no obvious physical reason for preferring it.
Turbulent flow that undergoes organized temporal unsteadiness is a subject of great importance to unsteady aerodynamic and thermodynamic devices. Among the classes of unsteady flows, those bounded by rigid smooth walls are particularly amenable to fundamental studies of unsteady turbulence and its modeling. The purpose of this article is to summarize recent contributions in this area, with a view to consolidating comprehension of the well-known basics of these flows, and drawing attention to critical gaps in information which restrict the understanding of unsteady turbulent flows.
Measurements on an unsteady turbulent boundary layer with well-defined
initial and boundary conditions are reported in this paper. The test
boundary layer is a standard, steady, flat-plate turbulent boundary
layer at the entrance to the unsteady region, and it is subjected to a
linearly decreasing free-stream velocity in a time-dependent manner. The
experiments were conducted in a specially designed water tunnel with the
frequency of the imposed oscillations ranging from quasi-steady (f + 0)
up to the 'bursting frequency' in turbulent boundary layers. It was
found that the mean velocity and the turbulence intensity profiles were
unaffected by the imposed oscillations. The amplitude ratio of the
periodic component, although as much as 1.7 for quasi-steady
oscillations, becomes unity over the outer region of the boundary layer
at higher frequencies. Both the boundary layer thickness and the
Reynolds stress distribution across the boundary layer become frozen
over the oscillation cycle at high frequencies.
The present paper is the first part of a two-part report on a detailed investigation of periodic turbulent pipe flow. In this investigation, experimental data on instantaneous velocity and wall shear stress were obtained at a mean Reynolds number of 50000 in a fully developed turbulent pipe flow in which the volumetric flow rate was varied sinusoidally with time around the mean. Two oscillation frequencies at significant levels of flow modulation were studied in detail. The higher of these frequencies was of the order of the turbulent bursting frequency in the flow, and the other can be regarded as an intermediate frequency at which the flow still departed significantly from quasi-steady behaviour. While a few similar experiments have been reported in the recent literature, the present study stands out from the others in respect of the flow regimes investigated, the magnitude of flow modulation, the detailed nature of the measurements and most importantly the identification of a relevant parameter to characterize unsteady shear flows. The present paper contains the main experimental results and comparisons of these results with the results of a numerical calculation procedure which employs a well-known quasi-steady turbulence closure model. The experimental data are used to study the manner in which the time-mean, the ensemble-averaged and the random flow properties are influenced by flow oscillation at moderate to high frequencies. In addition, the data are also used to bring out the capability and limitations of quasi-steady turbulence modelling in the prediction of unsteady shear flows. A further and more detailed analysis of the experimental data, results of some additional experiments and a discussion on the characterization of turbulent shear flows are provided in Part 2 (Ramaprian & Tu 1983).
Turbulent boundary layers with a time-dependent free-stream velocity
occur in many fluid dynamic devices. The main objective of the present
work are as follows: (1) to understand the characteristics of a
turbulent boundary layer that develops under steady, zero-pressure
gradient conditions upstream, and is subjected to well-defined and
sinusoidally varying adverse free-stream gradient downstream, and (2) to
provide a definitive data set that would serve as a test case for
evaluating the performance of calculation procedures and turbulence
models for such flows. The results show that the time-averaged velocity
and turbulence are insensitive to the imposed unsteadiness, while the
periodic components are very strongly affected by the frequency.
Calculations of separated flows with the low-Reynolds-number of q-[ turbulence model The Pennsylvania State University Calculation of impinging-jet heat transfer with the low-Reynolds-number q-[ turbulence model Computation of oscillat-ing turbulent flows at transitional Reynolds numbers
- M M Gibson
- R D Harper
- M M Gibson
- R D Harper
Gibson, M. M. and Harper, R. D. 1995. Calculations of separated flows with the low-Reynolds-number of q-[ turbulence model. Proc. lOth Turbulent Shear Flows Symposium, The Pennsylvania State University, University Park, PA Gibson, M. M. and Harper, R. D. 1996. Calculation of impinging-jet heat transfer with the low-Reynolds-number q-[ turbulence model. Proc. Int. Conf. Turbulent Heat Transfer, San Diego, CA Hanjalic, K., Jakirlic, S. and Hadzic, I. 1993. Computation of oscillat-ing turbulent flows at transitional Reynolds numbers. Proc. 9th Turbulent Shear Flows Symposium, Kyoto, Japan Huang, P. G. and Bradshaw, P. 1995. Law of the wall for turbulent flows in pressure gradients. A/AA J., 33, 624-632.