Quantification of velocity and pressure fields over streambeds is important for predicting sediment mobility, benthic and hyporheic habitat qualities, and hyporheic exchange. Here, we report the first experimental investigation of reconstructed water surface elevations and three‐dimensional time‐averaged velocity and pressure fields quantified with non‐invasive image techniques for a three‐dimensional free surface flow around a barely submerged vertical cylinder over a plane bed of coarse granular sediment in a full‐scale flume experiment. Stereo particle image velocimetry coupled with a refractive index‐matched fluid measured velocity data at multiple closely‐spaced parallel and aligned planes. The time‐averaged pressure field was reconstructed using the Rotating Parallel Ray Omni‐Directional integration method to integrate the pressure gradient terms obtained by the balance of all the Reynolds‐Averaged Navier‐Stokes equation terms, which were evaluated with stereo particle image velocimetry. The detailed pressure field allows deriving the water surface profile deformed by the cylinder and hyporheic flows induced by the cylinder.
A scaling patch approach is used to investigate the proper scales in planar turbulent wakes. A proper scale for the mean axial flow is the well-known maximum velocity deficit [Formula: see text], where [Formula: see text] is the free stream velocity and [Formula: see text] is the mean axial velocity at the wake centerline. From an admissible scaling of the mean continuity equation, a proper scale for the mean transverse flow is found as [Formula: see text], where [Formula: see text] is the growth rate of the wake width. From an admissible scaling of the mean momentum equation, a proper scale for the kinematic Reynolds shear stress is found as [Formula: see text], which is a mixed scale of the free stream velocity and the mean transverse flow scale. Expressions are derived for the scaled mean transverse velocity and Reynolds shear stress in the far field of planar turbulent wakes. Using a Gaussian function for the mean axial velocity deficit, approximate functions for the scaled mean transverse velocity and Reynolds shear stress are developed and found to agree well with experimental and simulation data. This work reveals that the mean transverse flow, despite its small magnitude, plays an important role in the scaling and understanding of the planar turbulent wake.
By using a combination of integral and self-similarity analyses, the generalized analytical solutions for the mean transverse velocity and Reynolds shear stress are rigorously derived for the first time for the far field of planar turbulent wakes under arbitrary pressure gradients. Specifically, by assuming self-similarity for the mean axial velocity, the analytical formulation for the mean transverse velocity is obtained from the integral of the mean continuity equation, and the analytical formulation for the Reynolds shear stress is obtained from the integral of the momentum equation. The generalized analytical formulations for the mean transverse velocity and Reynolds shear stress consist of multiple components, each with its unique scale and physical mechanism. In the zero pressure gradient limit, the generalized formulations recover the single-scale equations reported by Wei, Liu, and Livescu. Furthermore, simpler approximate formulations for the mean transverse velocity and Reynolds shear stress are also obtained, and show excellent agreement with the experimental measurements. The findings provide new insights into the properties of planar turbulent wakes under pressure gradients, filling some long-standing gaps in the existing literature.
Accurately and efficiently measuring the pressure field is of paramount importance in many fluid mechanics applications. The pressure gradient field of a fluid flow can be determined from the balance of the momentum equation based on the particle image velocimetry measurement of the flow kinematics, which renders the experimental evaluation of the material acceleration and the viscous stress terms possible. In this paper, we present a novel method of reconstructing the instantaneous pressure field from the error-embedded pressure gradient measurement data. This method utilized the Green's function of the Laplacian operator as the convolution kernel that relates pressure to the pressure gradient. A compatibility condition on the boundary offers equations to solve for the boundary pressure. This Green's function integral (GFI) method has a deep mathematical connection with the state-of-the-art omnidirectional integration (ODI) for pressure reconstruction. As mathematically equivalent to ODI in the limit of an infinite number of line integral paths, GFI spares the necessity of line integration along zigzag integral paths, rendering generalized implementation schemes for both two and three-dimensional problems with arbitrary inner and outer boundary geometries while bringing in improved computational simplicity. In the current work, GFI is applied to pressure reconstruction of simple canonical and isotropic turbulence flows embedded with error in two-dimensional and three-dimensional domains, respectively. Uncertainty quantification is performed by eigenanalysis of the GFI operator in domains with both simply and multiply connected shapes. The accuracy and the computational efficiency of GFI are evaluated and compared with ODI.
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