Turbulent oscillating channel flow subjected to a wind stress W. Kramer & H. J. H. Clercx1 Fluid Dynamics Laboratory2, Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands V. Armenio Dipartimento di Ingegneria Civile e Ambientale, Università degli studi di Trieste, Trieste, Italy ABSTRACT: Large eddy simulations of a periodic channel subjected to an oscillating pressure gradient and a wind stress at the free surface are presented. The resulting pulsating flow can be decomposed in a constant part which is the result of the wind stress and an oscillating part due to the pressure gradient. The pressure gradient first accelerates the flow leading to an increased wall stress and higher turbulence levels. Near the bottom a logarithmic layer is observed with strong turbulent streaks as is typical for steady boundary layer turbulence. When the flow is decelerated, streaks are weakened and smoothened by viscosity. After the flow reverses a log layer with turbulent streaks reappear when the wall stress builds up. At the free-surface the wind stress drives a constant production of turbulence and streaks are present throughout the cycle. As bottom and free-surface streaks are aligned mainly one-component turbulence is observed in the interior. 1 INTRODUCTION statistics in large-scale geophysical flows. To inquire the type of turbulence in such a geophysical flow we In addition to tidal forcing, the stress exerted on the performed large-eddy simulations of an oscillating free surface might play a role in driving the flow in flow subjected to a wind stress at the free surface. In estuary regions. Therefore, we estimate the ampli- the following sections we first describe the problem tude of both effects based on data for the Wester- and numerical model in Sec. 2 and present the re- schelde estuary. Here, the flow is strongly tidal sults in Sec. 3–5. A short discussion of the results driven with a typical fluid velocity during high tide and of future work is given in Sec. 6. of U = 0.2 – 1.0 m/s. The Reynolds number based 6 7 on the depth ( h = 10 m) is Re h = Uh /! = 10 – 10 2 PROBLEM DESCRIPTION with ! the kinematic viscosity. For this range of Re the flow is turbulent for all phases of the tidal cycle. In this work we study the turbulent oscillating chan- The estimated maximum wall stress due to the tides is 0.08–1 N/m2. A typical wind velocity of 7 m/s nel flow subjected to a wind stress by means of large yields a wind stress of 0.17 N/m2. The wind stress eddy simulations (LES). LES only resolves the larg- and the wall stress are thus of the same order. The est scales in the flow, while the effect of the smaller turbulent flow in the Westerschelde estuary is char- scales are modelled. The separation of the flow field acterized by a large Keulegan–Carpenter number, into large and small scales is achieved by filtering. KC = 100–600, which indicates that the time scale The velocity field ()()u,v, w = u1 ,u2 ,u3 can be writ- related to the turbulence is much shorter than the ten as u = uˆ + u"" with uˆ the filtered velocity and u"" tidal period. the small-scale part. Applying the filtering to the Turbulence is known to accelerate mixing and Navier–Stokes equation yields transport of particles. Hence, it is of relevance to sand sedimentation and dispersion of plankton for- uˆ #uˆ uˆ ˆ 2uˆ #% mations. This work is part of a project aimed at # i i j $1 #p # i ˆ ij + = $& +! + f i $ , (1) modeling dispersion of plankton in geophysical dt #x j #xi #x j #x j #x j flows. The approach is to first investigate particle dispersion at the smallest turbulent scales. This where & is the density of the fluid and pˆ the fil- knowledge can then be used for modeling dispersion tered pressure. The term % ij represents the stresses 1 Also at Department of Applied Mathematics, University of Twente, The Netherlands 2 Participates in J. M. Burgers Centre, Research school for Fluid Dynamics, The Netherlands acts on the free-surface layer. All quantities are made dimensionless using the height of the channel and the velocity amplitude of the tidal oscillation. Equation (1) is solved using a finite-volume method based on the method by Zang et al. (1994). The Reynolds number and Keulegan–Carpenter 4 number are decreased to Re h = Uh /! = 5!"# and KC = 80 to make the simulations feasible. As no wall-model is available for this kind of flow, the wall stress must be resolved by the LES. For the no- slip boundary layer to be resolved the first grid cell is set to be equal to one wall unit, which is defined + * * as (z = (z z with z =! u% and Figure 1. The domain describes a water column of size u . The maximum wall stress l × l × h bounded by a no-slip bottom and a free-surface % = % w,max & % w,max x y can be estimated in advance using the maximum at the top. An oscillating pressure gradient f p is applied over the x-direction, while a wind stress % wind is acting on wall stress for the purely oscillating case and adding the free surface in the same direction as the pressure gradi- the surface stress. Using data from Jensen et al. ent. (1989) this yields an estimated maximum of -3 % w,max = 3.0!"# . Resolving the free-surface layer the small scales of the flow exert on the large scales. proved to require a finer resolution. Here, the first As the small scales in the flow are not resolved in + grid cell height is (z = 1 2 . The horizontal grid LES a model is required for the subgrid stresses . % ij spacing required in the boundary layer for resolved In the simulations we use a dynamic eddy-viscosity + LES is (x ) 60 for the streamwise direction and model combined with a scale-similar model (Ar- + (y ) 30 for the spanwise direction. These re- menio & Piomelli, 2000). quirements are reached with a resolution of The channel domain is periodic along the hori- 48× 64 ×128 if grid stretching is applied for the ver- zontal x- and y-direction and is bounded in the verti- tical. For these grid resolutions the numerical model cal direction by a no-slip bottom ( z = 0 ) and a free- has been used successfully to simulate a turbulent surface layer at the top ( z = 1). For a sketch of the oscillating channel flow as is typical for the Gulf of domain see figure 1. The horizontal dimensions are Trieste (Salon et al., 2007). larger than the channel height to capture the largest eddies in the domain (lx × l y × h = 2×1.4 ×1). To mimic a tidal flow an oscillating pressure gradient 3 MEAN VELOCITY f p = $U' cos()'t (2) The obtained flow field is strongly turbulent during with frequency ' = 1/ 80 and velocity amplitude the complete cycle. The combination of the tidal and U = 1 is applied over the x-direction. Along the wind forcing results in a pulsating mean flow in the -3 positive x-direction a constant wind stress % wind =10 x-direction (figure 2). The mean flow u()z is ob- Figure 2. The mean streamwise velocity for oscillating channel flow subjected to a wind stress for different phases (a) 0°–60° (b) 90°–150° (c) 180°–240° and (d) 270°–330°. The darker colors represent the later phases, e.g. in (a) the profiles go from light gray, to dark gray to black for 0°, 30° and 60°, respectively. Figure 3. The phase-averaged stress at the wall. Integrated over a period the wall stress matches the wind stress. tained by plane and phase averaging. The wind stress drives a constant mean flow in the interior with strong shear layers at the bottom and free- surface layer. A similar velocity profile is observed, Figure 4. The mean streamwise velocity in wall units for (a) 0° (b) 90° (c) 180° and (d) 270°. The lines relate to the theo- although with a larger velocity, when the oscillating retical laws for the viscous sublayer and logarithmic layer. pressure gradient is absent. The pressure gradient will accelerate and decelerate the flow in positive x- ity is either small or reversing, no log layer is pre- direction in the first half period. While in the second sent. If the wall stress then increases again at the half period it will accelerate and decelerate the fluid phases 270°–300° a log layer reappears. in the negative x-direction. The mean velocity In the first half cycle the free surface layer does reaches a maximum velocity of about U = 1.4 in the not differ from the constant case, i.e. with only a interior at * = 100°. The flow is then decreases, wind stress acting on the free surface. Then subsequently reverses and reaches a value of U = - + + $1 + u $ u (z = h) = + log z + C describes the log 0.6. at * = 250°. layer at the free surface, but with + = 0.5 and C = 1. The wall stress is given in figure 3. Integrated The lower value of C relates to a thinner viscous over a period the wall stress matches the applied sublayer. Tsai et al. (2005) argued that presence of wind stress. An above-average wall stress is ob- horizontal fluctuations at the free-surface has a simi- served during a shorter period 20°–170° than a be- lar effect as surface roughness, which leads to a de- low average value.