Environ Fluid Mech (2008) 8:19–29 DOI 10.1007/s10652-008-9052-0 ORIGINAL PAPER
An engineering model for countercurrent flow under wind-induced waves and current Y. Yang · A. G. Straatman · H. Hangan · E. K. Yanful
Received: 7 June 2007 / Accepted: 5 January 2008 / Published online: 24 January 2008 © Springer Science+Business Media B.V. 2008
Abstract A general model for the phase-averaged velocity field in wind-induced countercurrent flow is proposed. The influence of waves on the time-averaged velocity is accounted for by introducing a skewness factor in a parabolic eddy viscosity model. The skewness factor represents the net effect of the wavy surface in the engineering model for velocity. The coherent velocity components are described separately by an orbital velocity obtained from linear wave theory and are added to the time-averaged components to give a complete model for the phase-averaged velocity field. The proposed model collapses to the standard model for deep-water conditions, but is also shown to yield the correct behavior for intermediate conditions. Moreover, the bed shear stress, derived from the proposed velocity model, is also shown to be in agreement with experiments. Keywords
Countercurrent flow · Shallow water · Semi-empirical model
1 Introduction Knowledge of the flow structure in shallow, closed water bodies and the mechanisms that drive the flow are of importance in pond ecosystem protection and management, water quality control, and tailings pond design, operation and maintenance. Generally, wind induces waves and drift on the upper surface of a body of water. The drift flow moves downwind near the
Y. Yang · H. Hangan · E. K. Yanful The Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 A. G. Straatman (B) The Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 e-mail:
[email protected] H. Hangan The Boundary Layer Wind Tunnel Laboratory, The University of Western Ontario, London, Ontario, Canada N6A 5B9
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upper surface due to surface shear, while a return or countercurrent flow is generated along the lower surface due to the positive pressure gradient that forms by wind set-up to satisfy mass conservation across a vertical plane in the closed water body. The influence of the waves on the submerged flow patterns can be classified broadly into three regimes, depending upon the wavelength to water depth ratio λ/ h: (a) deep-water conditions where waves have a minimal impact on the current flow (wave length λ less than twice the water depth h); (b) intermediate-water conditions where waves partially impact the current flow (2 < λ/ h < 20); and (c) shallow-water conditions where waves strongly influence the current flow (λ/ h ≥ 20). The wind-generated waves in typical closed water bodies are either deep or intermediate. For example, man-made shallow water covers used by the mining industry to isolate reactive mine tailings from the environment typically operate under intermediate-water conditions. Shallow-water conditions seldom occur, except near beach areas, since the wavelength rarely exceeds 20 times the water depth. The submerged flow motions are also classified as laminar, transitional or turbulent, depending upon the surface Reynolds number Re = u s h/ν, based on water depth, h, kinematic viscosity of water, ν, and surface velocity, u s . To this end, Bye [2] numerically estimated the critical Reynolds number for initial loss of hydrodynamic stability in closed basins of infinite fetch to be approximately Re = 500 ± 100, which is also close to that predicted by Ramanan and Homsy [11] for lid-driven square cavities. Generally, with increasing surface velocity, the flow evolves and eventually undergoes transition to fully turbulent flow, which was first observed in a square cavity at Re = 6,000–8,000 [7]. For typical wind–water interactions, the surface Reynolds number exceeds 10,000 and thus the flow field is normally fully turbulent. The flow structure in closed bodies of water subjected to surface shear and waves remains poorly understood. Most early efforts to investigate such flows focused solely on the surface drift driven by wind and waves. Laboratory studies by Wu [18–20], Plate [10] and Spillane and Hess [13] made progress in the understanding of wind-induced surface drift in confined water bodies. These studies focused on the wind-induced surface motions under fairly small wind generated waves (deep-water conditions). However, no attention was directed towards the underlying flow structures in the water. Later, Koutitas and O’Connor [8], Goossens et al. [3], Tsuruya et al. [17], Tsanis and Leutheusser [15], Tsanis [16] and Wu [21] made progress in examining the countercurrent (water) flow induced by pure surface shear stress (without waves). Tsanis [16] proposed a double logarithmic law for the mean velocity field for this countercurrent flow, which was then improved by Wu and Tsanis [21]. The model was based on the flow motion driven by a flat, translating wall and therefore free surface effects were not accounted for. The double log-law was derived by assuming a parabolic distribution of eddy viscosity in the vertical plane and a linear shear stress distribution, and since surface effects were not accounted for, is strictly only valid for deep-water wave conditions. No attempt was made to generalize the model to account for the surface layer where the influence of waves is significant and thus to make the model valid for intermediate and shallow conditions. The simulation of wind-driven waves and currents in bodies of water has also long been a challenge for computational fluid dynamics (CFD) because of the complicated physics at the air–water interface. Groeneweg and Klopman [4] made progress towards the solution for wind-wave interaction in open bodies of water using a generalized lagrangian mean (GLM) approach. The formulation was initiated from the simplified mass and momentum conservation equations and included a one-equation turbulence model to compute the vertical eddy-viscosity distribution. Unique boundary conditions were derived to account for the wavy surface, but comparisons to related experiments indicated that the solution was only accurate for weakly convective flows. Efforts by Belcher et al. [1], Harris et al. [5] and
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Sullivan et al. [14] have made progress in understanding wind and wave induced flow structures in closed water bodies using experiments, analysis and direct numerical simulations (DNS). Formulation of the theoretical problem was also based on a solution of the simplified mass and momentum equations, but the wave shape was imposed using a time dependent function and not obtained by a coupled solution of the air–water interface. While it is of practical interest to use CFD tools in the design and maintenance of shallow water bodies, complex bed topology, wind–water interaction involving shear and wave setup, and scale are just a few of many difficulties associated with this application. An alternate approach to studying such applications is to attempt to reduce the problem by one dimension, similar to the study of thin fluid layers. By this approach, a two-dimensional problem would be solved computationally while integrating the effect of the third dimension into the equation set. In this approach, an engineering model is required to characterize the velocity profile in the reduced dimension. The focus of the present work is to develop a model that characterizes countercurrent flow in closed, shallow water bodies driven by wind and wave motions that is valid over the full spectrum of λ/ h. The immediate need for a model is in the characterization of existing shallow water systems and as a design tool for the development of new shallow water facilities for the mining industry. The emphasis on shallow water in this particular application is based on the fact that a deep water cover is not an attractive option for the mining industry because of the need to construct high storage dams, which tend to compromise geotechnical stability. The resulting countercurrent flow model may also be used in the development of an alternate approach for CFD modeling, as described above. This paper is organized such that the model formulation for the velocity and bed shear stress is presented first, followed by validation with available experimental data.
2 Model formulation Velocity profiles and bed shear stresses under deep-water conditions are well-described using the turbulent countercurrent model developed by Tsanis [16] and Wu and Tsanis [21], since the influence of surface waves is insignificant in most of the flow field. Under intermediate and shallow water conditions, however, where the flow structure is (strongly) influenced by the surface waves, the Wu and Tsanis [21] model does not accurately describe the resulting velocity field. To describe the velocity and stress profiles under intermediate and shallow wave conditions, a more general model is needed. Thus, a model of the form u(z, t) = u(z) + u c (z, t) w(z, t) = wc (z, t)
(1)
is proposed, where u(z) ¯ is the time-averaged velocity, u c (z, t) and wc (z, t) are the organized coherent components, z is the vertical direction, and the brackets indicate that the modeled velocity field represents the phase-average. The phase-average implies that the model describes one cycle of a quasi-periodic flow field. The fact that the model is cast in a phaseaveraged form suggests that a dominant frequency exists around which the phase-average can be constructed. In the present formulation, it is assumed that the dominant wave-passing frequency is suitable to form the phase average and the rest of the spectrum will be accounted for in the remaining random turbulent terms. This approach is, therefore devised based on the well-known triple-decomposition approach proposed by Hussain and Hayakawa [6].
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The present study focuses on a two-dimensional fully-developed countercurrent flow region. The domain of interest for the proposed model is shown in Fig. 1. It is essentially a channel-like domain with a flat lower surface (z = 0) and a flat upper surface (z = h) located at the mean water depth level, i.e. the waves are not considered explicitly; only the mean water depth. The flow is assumed fully-developed in the streamwise, x, direction. Models for the time-averaged and coherent components are devised separately on the assumption that there is sufficient separation between the scales of the random turbulence and the scale of the (dominant) coherent wave motions. Derivation of the two components of the model follows. 2.1 Time-averaged component Assuming an incompressible, Newtonian fluid, the simplified, time-averaged conservation of mass and momentum equations are: ∂w =0 ∂z 1 ∂ p¯ ∂u ∂ − + νeff =0 ρ ∂x ∂z ∂z 1 ∂ p¯ =0 − ρ ∂z
(2)
in which u¯ and w¯ are the time-averaged velocity components in the x and z directions, respectively, p¯ is the time-averaged pressure and νeff (= ν + νt ) is the effective eddy viscosity. No-slip, zero-penetration conditions are imposed on the lower surface while the upper surface (mean depth level) has an induced stress given by τs = ρu 2s ∗ (where u s ∗ is the surface friction velocity) and a vertical velocity, w¯ = 0. Because the water body is considered closed, the flow is subject to the additional constraint that the integrated mass flux along any vertical cross-section is zero: h u¯ dz = 0
(3)
0
Equations 1–3 along with the stated boundary conditions represent the complete problem to be solved for the time-averaged velocity component. To obtain a solution for the velocity, an expression for the turbulent eddy viscosity is required. Tsanis [16] suggested a parabolic expression of the form: νt =
u ∗s (z + z b )(z s + h − z) h
Fig. 1 Schematic of geometry under consideration
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(4)
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on the assumption that a linear shear stress distribution exists. In Eq. 4, z b and z s are characteristic heights at the bottom (z = 0) and the surface of water (z = h), respectively, and are used to characterize the viscous sub-layers at the bottom and surface of the water. is a constant used to characterize the intensity of the turbulence. According to Tsanis [16] and Wu and Tsanis [21], z s / h = 2.2 × 10−4 and z b / h = 1.4 × 10−4 for fully-turbulent countercurrent flow conditions. In addition, was estimated to be between 0.2 and 0.5 with an average value of about 0.35 for surface Reynolds numbers (Re) ranging from 103 to 105 [16]. When Eq. 4 is combined with Eqs. 2 and 3 and solved, the result is a velocity profile that is accurate for fully-turbulent deep-water conditions. A salient feature of this velocity distribution is that the zero-crossing is at z/ h = 2/3, independent of the values of the parameters used in the model. While this is accurate for deep-water conditions, the recent measurements of Yang [22] suggest that for intermediate-water conditions, the shear stress distribution is not linear, and the zero-crossing is nearer to the (mean) water surface. A straightforward way to account for the non-linear shear stress profile and allow for a change in the position of the zero-crossing is to modify Eq. 4 as: u ∗s z νt = , (5) (z + z b ) (z s + h − z) 1 + a h h where the coefficient a is a skewness factor for the eddy viscosity profile. This skewness factor should not be thought to represent a specific physical process, but rather to account for the net effect of the (wavy) upper surface in the engineering model. The notion that the upper and lower layers of a wind-wave driven flow should have a different impact on the resulting shear stress profile is not new. In a simplified model for wind-driven ocean currents, Madsen [9] proposed a double-linear model to account for the different effects of the bottom and wavy surfaces of the ocean, to avoid a non-physical linear shear stress profile. In the present case, the skewess factor simply changes the influence of the upper and lower layers enabling the prediction of more physical shear stress and velocity profiles. Using Eq. 5 to approximate the eddy viscosity and applying the boundary conditions and the zero mass-flux constraint, the following solution for the time-averaged component of Eq. 1 emerges: u z z z , (6) = A ln 1 + + C ln 1 + a + B ln 1 − u ∗s zb zs + h h where, A= B= C =
−(1 + a) ln(1 + a) + az sh ln 1 +
1 z sh
D (1 + a) ln(1 + a) − a(1 + z bh ) ln 1 + a(1 + z bh ) ln 1 +
D
1
z bh
− az sh ln 1 +
1 z bh 1 z sh
D 1 D = a [1 + a(1 + z sh )] (1 + z bh )2 ln 1 + z bh 1 2 + az sh (1 − z bh ) ln 1 + z sh − (1 + z bh + z sh )(1 + a)2 ln (1 + a)
(7)
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2.2 Coherent component The coherent components in Eq. 1 are devised by considering the wave-induced orbital velocities based on linear wave theory. The horizontal and vertical velocity components of the water particle velocity in a wave can be obtained from the potential flow assumption [12] and are: u(x, z, t) =
π H cosh(kz) cos(kx − σ t) T sinh(kh)
(8)
w(x, z, t) =
π H sinh(kz) sin(kx − σ t) T sinh(kh)
(9)
where H is the wave height, T the wave period, k = 2π/λ the wave number, λ the wavelength, σ = 2π/T the wave angular frequency, and t is time. It was assumed from the outset that the scales of the turbulent and coherent motions were significantly different, and in essence, that the coherent motions are largely unaffected by the turbulence in the flow. To demonstrate the validity of the assumptions of scale separation and linear wave theory, Fig. 2 gives a comparison of the measured coherent velocity components u c and wc [22] with the wave orbital velocity obtained from Eqs. 8 and 9 in a period for a point very near the surface of the water. Figure 3 gives the maximum coherent velocity with the maximum wave orbital velocity in the vertical plane, again compared with Eqs. 8 and 9. Both figures show very good agreement in most parts of the water body and in time. In particular, the predictions for the vertical component are in excellent agreement with LDV measurements. Some scatter is observed for the shear regions near the surface and floor of the tank for the longitudinal component. The complete model for u(z, t) and w(z, t) that accounts for the influence of induced surface shear and waves is formed by substituting models for the time-averaged and coherent components into Eq. 1. The resulting model expressions are:
u(x, z, t) z z z + C ln 1 + a = A ln 1 + + B ln 1 − u ∗s zb zs + h h πH + T u ∗s
time averaged term
cosh(kz) cos(kx − σ t) 0 ≤ z ≤ h sinh(kh)
(10)
coherent velocity term
w(x, z, t) π H sinh(kz) sin(kx − σ t) 0 ≤ z ≤ h = u ∗z T u ∗ sinh(kh) s
(11)
coherent velocity term
where A, B, C and D are given in Eq. 7. The proposed engineering model was tested for several conditions and it was found to be suitable to describe the phase-averaged velocity profiles under any wave condition. 2.3 Engineering model for bed shear stress The model is extended to give an estimate of the bed shear stress, which is useful for determining the onset of bed particle motion and hence suspension into the water cover. The bed shear stress is obtained by simply taking the first derivative of Eq. 10, multiplying by the eddy viscosity, and setting z = 0:
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Fig. 2 Comparison of the coherent components of the proposed model to experimental data from Yang [22] for an intermediate-water condition
τ (x, z, t) =
ρu 2∗s λ ⎛h
(z b ) (z s + 1)
⎞
⎟ ⎜ A π H sinh(kz) B C ⎟ ⎜ + k − +a cos(kx − σ t)⎟ ·⎜ ⎠ ⎝ zb / h zs / h + 1 1 + az/ h T u ∗s sinh(k) Coherent component Time-averaged component (12) For deep-water conditions, the skewness factor, a, is zero and the wave period T ⇒ ∞, yielding a constant value for the bed shear stress. For intermediate-water conditions, the
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Fig. 3 Coherent components of u and w velocity along vertical plane in the wave tank compared with particle orbital velocity predicted by linear wave theory for one intermediate wave condition considered by Yang [22]
skewness factor and the wave period take their established values, but z is only set to zero in the time-averaged component. In the coherent component, z must be set to a characteristic height above the bed to yield the correct magnitude of the periodic bed shear stress. Comparison to bed shear stress values extracted from the present experiments suggests a value z/ h = 0.08. Proper calibration of this characteristic height requires further experiments under many different water-wave conditions. The bed shear stress is also subject to a random component, which could be evaluated based on the standard deviation of the turbulence field. As such, the model could be used to give a conservative estimate of the maximum bed shear stress, which is often required when designing shallow water covers for mine tailings storage. 3 Validations The proposed model is compared to results for deep and intermediate water-wave conditions obtained from a set of wind-wave tank experiments conducted by Yang [22]. Figure 4 shows a
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comparison of the time-averaged velocity u with Wu and Tsanis [21] model and experiments from Yang [22] for deep water-wave conditions. The figure shows that the deep water-wave result is described accurately and that the proposed model collapses to the result of Tsanis model for a = 0, as required. It is also worth noting that the zero crossing for both sets of predictions shown in Fig. 4 occurs at 2/3h, which is in accordance with the deep water-wave data included in the figure. Figure 5 gives a comparison of the complete model to experimental data for the minimum, mean and maximum velocities in one wave period for an intermediate water-wave condition considered by Yang [22]. The parameters used in the model are the same as those suggested by Tsanis, with the exception of a, which was adjusted to give the best fit to the data. The present model is shown to be in good agreement in terms of shape and magnitude, thereby justifying the modeling approach used. Most important is the fact that the zero-crossing is seen to move along the z-axis; though the model predictions do not precisely match the measured data. With further adjustment, a closer fit could be obtained. It is important to note that the proposed engineering model is shown here to be capable of representing cases of wind/wave-driven countercurrent flow. Further study is required to determine if the model parameters are universal or if they require adjustment for different flow conditions. Figure 6 gives a comparison of the phase-averaged bed shear stress compared to Eq. 12 (using z/ h = 0.08 in the coherent component). Again, the comparison indicates that for the chosen value of z/ h, the shear stress model is capable of accurately describing the phaseaveraged bed shear stress. Fig. 4 Comparison of the time-averaged component of the proposed model to experimental data for a deep-water condition. Included for comparison is the model of Wu and Tsanis [21], which is only valid for deep-water conditions
Fig. 5 Comparison of the modeled results and measured data for three instants in one phase of a wave cycle for an intermediate-water condition
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Fig. 6 Comparison of predicted and measured bed shear stress under one intermediate water condition considered by Yang [22]
4 Summary A general engineering model for the velocity distribution in wind-induced countercurrent flow was developed taking into consideration the nonlinear shear stress profile and the phaseaveraged velocity field. The influence of waves on the time-averaged velocity was accounted for by introducing a skewness factor in the eddy viscosity model of Tsanis [16]. The skewness factor is not thought to represent any specific physical process, but rather to represent the net effect of the surface in the engineering model. Comparisons to the present experiments showed excellent agreement with both the time-averaged and phase-averaged velocities. An estimate of the bed shear stress derived from the model was also in good agreement with the present experimental data. While there is no inherent limitation in the model, further validation is needed to assess the proper values of the coefficients and to determine the potential universality of the coefficients. Acknowledgements The Natural Sciences and Engineering Research Council of Canada (NSERC) and an industry consortium consisting of Rio Algom, Noranda, Falconbridge, Inco, Teck, Cambior, and Battle Mountain supported this research. Special thanks also go to Brian Havel, who helped in collecting data using the LDV.
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