We present a simple viscous theory of free-surface flows in boundary layers, which can accommodate regions of separated flow. In particular this yields the structure of stationary hydraulic jumps, both in their circular and linear versions, as well as structures moving with a constant speed. Finally we show how the fundamental hydraulic concepts of subcritical and supercritical flow, originating from inviscid theory, emerge at intermediate length scales in our model.
On the basis of the viscous Saint-Venant equations, hydraulic jumps in laminar open channel flow are obtained as continuous shock structures. Thanks to the inclusion of viscosity, the jumps are not abrupt, rendering the classic patchwork via the Rankine-Hugoniot shock relations unnecessary. The jumps arise as stable stationary solutions of the governing equations and lend themselves excellently to a Dynamical Systems analysis, manifesting themselves as near-parabolic trajectories in phase space. Based on this, we derive an analytic expression for the jump length as a function of the Froude and…
This paper gives a new unified formula for the Newtonian fluids valid for all pipe flow regimes from laminar to the fully rough turbulent. It includes laminar, unstable sharp jump from laminar to turbulent, and all types of the turbulent regimes: smooth turbulent regime, partial non-fully developed turbulent and fully developed rough turbulent regime. The formula follows the inflectional form of curves as suggested in Nikuradse's experiment rather than monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula consists of switching functions and of the int…
We carry out an analytical study of laminar circular hydraulic jumps, in generalized-Newtonian fluids obeying the two-parametric power-law model of Ostwald-de Waele. Under the boundary-layer approximation we obtained exact expressions determining the flow, an implicit relation for the jump radius is derived. Corresponding results for Newtonian fluids can be retrieved as a limiting case for the flow behavior index n=1, predictions are made for fluids deviating from Newtonian behavior.
The formation of a planar hydraulic jump has been analysed in the framework of a full depth-averaged thin film model (DAM) with surface tension effects included. We have demonstrated regular weak solutions of the full DAM and analysed surface tension effects. It has been shown that surface tension effects within the parameter range relevant to the recent experiments are expected to be very weak and practically negligible. The developed methodology can be used in the analysis of laminar flow regimes and as a benchmark in developing full scale hydrodynamic models.
The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously-forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms with (i) recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu & Atoufi et al., arXiv:2301.09773, 2023), (ii)…
High flow rate, water-driven hydraulic fractures are more common now than ever in the oil and gas industry. Although the fractures are small, the high injection rate and low viscosity of the water, lead to high Reynolds numbers and potentially turbulence in the fracture. Here we present a semi-analytical solution for a blade-shaped (PKN) geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. We model the turbulence in the PKN fracture using the Gaukler-Manning-Strickler parametrization, which relates the the flow rate of the water to the p…
Heat and fluid flow in a rectangular channel heat sink equipped with longitudinal vortex generators have been numerically investigated in the range of Reynolds numbers between 25 and 200. Aqueous solutions of carboxymethyl cellulose (CMC) with different concentrations (200-2000 ppm), which are shear-thinning non-Newtonian liquids, have been utilised as working fluid. Three-dimensional simulations have been performed on a plain channel and a channel with five pairs of vortex generators. The channels have a hydraulic diameter of 8 mm and are heated by constant wall temperature. The vortex genera…
In the presence of viscosity the hydraulic jump in one dimension is seen to be a first-order transition. A scaling relation for the position of the jump has been determined by applying an averaging technique on the stationary hydrodynamic equations. This gives a linear height profile before the jump, as well as a clear dependence of the magnitude of the jump on the outer boundary condition. The importance of viscosity in the jump formation has been convincingly established, and its physical basis has been understood by a time-dependent analysis of the flow equations. In doing so, a very close …
We study heat transfer in plane Couette flow laden with rigid spherical particles by means of direct numerical simulations using a direct-forcing immersed boundary method to account for the dispersed phase. A volume of fluid approach is employed to solve the temperature field inside and outside of the particles. We focus on the variation of the heat transfer with the particle Reynolds number, total volume fraction (number of particles) and the ratio between the particle and fluid thermal diffusivity, quantified in terms of an effective suspension diffusivity. We show that, when inertia at the …
In this paper we re-examine the flow produced by the normal impact of a laminar liquid jet onto an infinite plane when the flow is dominated by surface tension. It is observed experimentally that after impact the liquid spreads radially over the plane away from the point of impact in a thin film. It is also observed that, at a finite radius, there is an abrupt increase in thickness of the film which has been identified as a hydraulic jump,and that this radius is independent of the orientation of the surface showing that gravity is unimportant (Bhagat et al. 2018). We show that the application …
We study laminar thin film flows with large distortions in the free surface using the method of averaging across the flow. Two concrete problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile like in Karman-Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump where previous studies encountered a singularity. Solutions of the system are in …
Pore-scale observation of vortical flow structures in porous media is a significant challenge in many natural and industrial systems. Vortical structure dynamics is believed to be the driving mechanism in the transition regime in porous media based on the pore Reynolds number, $Re_p$. To examine this assertion, a refractive-index matched randomly packed porous medium is designed to measure the scale of vortical flow structures in transition from unsteady laminar to turbulent using two-dimensional time-resolved Particle Image Velocimetry (PIV). Planar PIV data for $Re_p$ from 100 to 948 is used…
Rigorous vertical averaging for pseudo-3D modeling of precipitation obstructed fluid flow
University of Stuttgart（斯图加特大学）；Stuttgart Center for Simulation Science, University of Stuttgart（斯图加特大学斯图加特模拟科学中心）；Helmholtz Centre for Environmental Research（亥姆霍兹环境研究中心）；Department Technical Biogeochemistry, Helmholtz Centre for Environmental Research（亥姆霍兹环境研究中心技术生物地球化学系）；Department of Chemical and Biological Engineering, Center for Biofilm Engineering, Montana State University（蒙大拿州立大学生物膜工程中心化学与生物工程学系）；Department of Hydromechanics and Modelling of Hydrosystems, University of Stuttgart（斯图加特大学水力学及水系统建模系）
Shallow flow models are used for a large number of applications including weather forecasting, open channel hydraulics and simulation-based natural hazard assessment. In these applications the shallowness of the process motivates depth-averaging. While the shallow flow formulation is advantageous in terms of computational efficiency, it also comes at the price of losing vertical information such as the flow's velocity profile. This gives rise to a model error, which limits the shallow flow model's predictive power and is often not explicitly quantifiable. We propose the use of vertical moments…
