Supercritical Free Surface Flow Research Articles

Computation of 2D Supercritical Free-Surface Flow in Rectangular Weak Channel Bends

Computation of 2D Supercritical Free-Surface Flow in Rectangular Weak Channel Bends

Analytical solution for supercritical flow in rectangular channel contraction

Abstract Analysis of the supercritical flow pattern in nonprismatic open channels is an important issue in engineering practice and often a challenging question principally due to the oblique standing wave structure. In this paper, an original analytical solution for supercritical free-surface flow in rectangular channel contraction is proposed. The two-dimensional steady depth-averaged shallow water equations are simplified to an asymptotic one-dimensional (1D) unsteady flow problem in the transverse direction. A first-order approximation is then obtained using small perturbation theory, and the 1D asymptotic model is solved analytically using the Laplace integral transformation. The two-dimensional flow field solution is obtained following a compilation of sequence planes in the time domain. Free-surface profiles along the side wall and the channel axis were compared against experimental data. For this, laboratory tests were performed in addition to a comparison with classical experimental results available in the literature. Within the limits underlying the model, good agreement in water surface profiles is obtained by comparing experimental and theoretical predictions. Principal flow features, such as wave amplitude and positions, are well captured, which makes the solution model of interest for the design and analysis of contracted open channels.

Large-eddy simulation of supercritical free-surface flow in an open-channel contraction

Large-eddy simulations (LES) of supercritical flow in a straight-wall, open-channel contraction of 6° and contraction ratio of 2:1 are performed. The LES code solves the filtered Navier-Stokes equations for two-phase flows (water-air) and employs the level-set method. The simulation was validated by replicating a previously reported experiment. Contours of the time-averaged velocities indicate that the flow loses energy and momentum in the contracting channel. Further, secondary currents in the contraction are redistributing momentum and are responsible for local up-and down-flows. The turbulent kinetic energy reaches very high values at the entrance of the contraction, mainly contributed by the streamwise normal stress. The flow contains coherent turbulence structures which are responsible for carrying low-momentum from the bed and the water surface towards the channel centre. Flow deceleration results in significant turbulence anisotropy in the contracted section. It is shown that mainly pressure drag contributes to the energy loss in the contraction.

Solvability of a supercritical free surface flow problem under gravity-capillarity

In this paper, we consider a problem of a supercritical free surface flow over an obstacle lying on the bottom of a channel in 2D. The flow is irrotational, stationary and the fluid is ideal and incompressible. We take into account both the gravity and the effects of the superficial tension. The problem is nonlinear, it is formulated by the Laplace operator and the dynamic condition defined on the free surface of the fluid domain (Bernoulli equation). Using the perturbation stream function, we linearize the problem and we give a priori properties of the solution. These a priori properties allow us to construct a space where we can use the Lax–Milgram’s theorem to prove the existence and the uniqueness of the solution of the problem.

Quasi-2D model for computation of supercritical free surface flow in sudden expansion

This work deals with the mathematical modeling of a supercritical flow at an abrupt channel expansion. A simplified approach for the study of the characteristics and features of the flow in the presence of the cross shock waves is then proposed. Based on the two-dimensional Saint-Venant equations and using certain simplifying assumptions in practice, this 2D shallow water model is reconverted to an equivalent 1D flow problem by inverting space time variables. The numerical solution of the 1D equation then makes it possible to reconstitute the solution of the 2D flow field according to translating planes. The results obtained show a satisfactory agreement with the experiments carried out for this problem. The profiles of the waterline along the wall and the flow axis are accurately reconstituted thus allowing the good design of the channel walls and the positioning of the oblique shock waves.

Discharge Coefficient of Rectangular Side Weirs on Circular Channels

AbstractIn this study, the flow turbulence and variations of the supercritical free surface flow in a circular channel along a side weir are simulated as three dimensional using the RNGk-εturbulence model and volume of fluid (VOF) scheme. Comparison between the numerical model and experimental measurements shows that the numerical model simulates the free surface flow with good accuracy. According to the numerical model results, the specific energy variations along the side weir for the supercritical flow regime are almost constant and the energy drop is not significant but by increasing the side weir length the energy difference between the side weir upstream and downstream increases. Next, using the nonlinear regression (NLR) and analysis of the simulation results, some relationships for calculating the discharge coefficient of side weir on circular channels in supercritical flow regime are provided.

Considerations on shock wave/boundary layer interaction in undular hydraulic jumps in horizontal channels with a very high aspect ratio

It can be seen in the literature that the fundamental factors governing oblique shock wave development, typically in very large channels with straight sidewalls, have not yet been completely understood and remain at the level of indicating its presence and formation. In this study, in addition to an analysis of various properties of hydraulic jump behaviour in very large channels, some aspects of boundary layer development and its detachment from the channel lateral sidewall are also investigated. At the detachment point of the lateral shock waves, it was noted that the displacement thickness experiences a significant increase; this is accompanied by a significantly reduced gradient normal to the channel sidewalls of the flow velocity as well as the occurrence of a strong, sudden adverse pressure gradient. An analysis of the flow velocity distribution and the background turbulence intensity of both the streamwise and spanwise velocity components was also carried out. Furthermore, it is argued that the supersonic flow separation analogy with a supercritical free surface flow can be applied to this case study and that the behaviour of the supercritical flow during separation can be interpreted by the free interaction theory typically used in aerodynamics.

Identifiability and Stability of Boundaries in a Supercritical Free Surface Flow

In this paper, we have studied a problem of identifiability of boundaries and stability of the solutions for the direct and the inverse problem concerning a supercritical and irrotational ow of an inviscid uid over an obstacle which lies on the bottom of a channel. The identifiability of the solution means its uniqueness when it exists. The stability is studied in the sense that for the direct problem and the inverse one, we study the variation of the obtained geometry for a little perturbation of the bottom or of the free surface. The proofs of the theorems are based on Holmgren theorem and the mean value theorem. The stability obtained is linear.

Bow Wave Dynamics

Experimental studies of air entrainment by breaking waves are essential for advancing the understanding of these flows and creating valid models. The present study used experimental simulations of a ship bow wave to examine its dynamics and air entrainment processes. The simulated waves were created by a deflecting plate mounted at an angle in a supercritical free-surface flow in a flume. Measurements of the bow wave geometry at two scales and also for a bow wave created by a wedge in a towing tank are presented. Contact line and bow wave profile measurements from the different experiments are compared and demonstrate the similarity of the flume simulations to the towing tank experiments. The bow wave profile data from the towing tank experiments were used to investigate the scaling of the wave with the flow and the dependence on geometric parameters. In addition, surface disturbances observed on the plunging wave are documented herein because of the role they play in air entrainment. The air entrainment itself is explored in Waniewski et al (2001).

Implicit Bidiagonal Scheme for Depth-Averaged Free-Surface Flow Equations

A general fast implicit bidiagonal numerical scheme, based on the MacCormack's predictor-corrector technique requiring the inversion of only block bidiagonal matrices, has been developed and subsequently applied for subcritical and supercritical free-surface flow calculations. The model has been applied to depth-averaged steady flows. There are two main advantages of the proposed method: the technique has fast convergence and utilizes a body fitted nonorthogonal local coordinate system to simulate irregular geometry flows. The model is used to analyze a wide variety of hydraulic engineering problems including flows in a converging-diverging subcritical flume, supercritical expansions at various Froude numbers, and supercritical converging chutes. For each of these test cases, the calculated results are compared with experimental data. The comparisons with measurements as well as with other numerical solutions show that the proposed method is comparatively accurate, fast, and reliable.

Unconfined Expansion of Supercritical Water Flow

The spread of supercritical free-surface flow on a smooth horizontal plane is considered. Experiments with selected approach Froude numbers are presented indicating that the effect of the Froude number may be dropped for hypercritical flow, that is, for values larger than 3. Also, a simple relation between local streamline direction and local flow depth is established. For sufficiently large approach flow depth, the Froude similarity law governs the phenomenon. Assuming that the streamwise velocity component is constant yields a system of equations identical to the one-dimensional simple wave problem. The solutions are compared with observations, and reasonable agreement is noted. Further particularities of hypercritical channel flow are established that are important for the numerical simulation of such currents. The features of expansion flow are documented with selected photographs. Supercritical unconfined expansion flow on a horizontal plane is studied. The governing equations can be shown to simplify considerably for hypercritical flow. A complete description is given based on both computations and experiments.

Supercritical free-surface flow with a stagnation point due to a submerged source

Two-dimensional free-surface flow produced by a submerged source in a fluid of finite depth is considered. It is assumed that there is a stagnation point on the free surface immediately above the source. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for various values of the Froude number F. It is found that there is a flow for each value of F ≥1.22. In addition, a local solution is constructed to describe the flow near the stagnation point as F→∞.

Symmetric supercritical free surface flow over a polygonal obstacle

In this paper, we consider steady, supercritical, plane, incompressible and irrotational flow in a channel with an obstacle of polygonal shape placed on the bottom. The solution is based on the combined use of a conformal mapping and the Riemann-Hilbert solution of a mixed boundary-value problem in the upper half plane. Numerical methods are introduced, and numerical solutions for some symmetric examples are presented. Numerical checks are applied showing good agreement. A comparison with the solution of the corresponding linearised problem shows that there are large differences, and contrary behaviour of the shape of the free surface as the Froude number is varied.