Impact of a rising stream on a horizontal plate of finite extent

2009 ◽  
Vol 621 ◽  
pp. 243-258 ◽  
Author(s):  
P. CHRISTODOULIDES ◽  
F. DIAS
Keyword(s):  
Galerkin Method ◽  
Froude Number ◽  
Steady Flow ◽  
Stagnation Point ◽  
Critical Value ◽  
Horizontal Plate ◽  
Free Surfaces ◽  
Bernoulli’S Equation ◽  
Asymmetric Flows ◽  
Finite Extent

The steady flow of a stream emerging from a nozzle, hitting a horizontal plate and falling under gravity is considered. Depending on the length of the plate L and the Froude number F, the plate can either divert the stream or lead to its detachment. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and L. The lift exerted on the plate is computed and discussed. If the plate is long enough, physically meaningful solutions are found to exist only for values of F greater than or equal to a certain critical value F0, which is to be determined. Results are presented, both for F > F0 where the detachment is horizontal and for F = F0 where the detachment point is a stagnation point at a 120° corner. Related asymmetric flows where the rising stream is inclined are also studied.

Impact of a falling jet

2010 ◽  
Vol 657 ◽  
pp. 22-35 ◽  
Author(s):  
PAUL CHRISTODOULIDES ◽  
FRÉDÉRIC DIAS
Keyword(s):  
Galerkin Method ◽  
Froude Number ◽  
Steady Flow ◽  
Maximum Pressure ◽  
Horizontal Plate ◽  
Vertical Pipe ◽  
Simplified Models ◽  
Free Surfaces ◽  
Bernoulli’S Equation ◽  
The Impact

Given the complexity of the problem of the impact of a mass of liquid on a solid structure, various simplified models have been introduced in order to obtain some insight on particular aspects of the problem. Here the steady flow of a jet falling from a vertical pipe, hitting a horizontal plate and flowing sideways is considered. Depending on the elevation H of the pipe relative to the horizontal plate and the Froude number F, the flow can either leave the pipe tangentially or detach from the edge of the pipe. When the flow leaves tangentially, it can either be diverted immediately by the plate or experience squeezing before being diverted. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and H. The pressure exerted on the plate is computed and discussed. For a fixed value of F, the maximum pressure along the plate goes through a minimum as H increases from small to large values. Results are presented for the three possible configurations: (i) tangential departure from the pipe and no squeezing, (ii) tangential departure from the pipe followed by squeezing of the liquid and (iii) detachment of the liquid from the pipe (with subsequent squeezing).

scholarly journals A note on the flow induced by a line sink beneath a free surface

1991 ◽  
Vol 32 (3) ◽  
pp. 251-260 ◽  
Author(s):  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Ideal Fluid ◽  
Critical Value ◽  
Homogeneous Fluid ◽  
Substitution Method ◽  
A Value ◽  
Line Sink ◽  
Low Froude Number

AbstractThe problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

Oscillatory flow past a circular cylinder in a rotating frame

1997 ◽  
Vol 345 ◽  
pp. 101-131
Author(s):  
M. D. KUNKA ◽  
M. R. FOSTER
Keyword(s):  
Circular Cylinder ◽  
Steady Flow ◽  
Stagnation Point ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Temporal Integration ◽  
Oncoming Flow ◽  
Flow Past ◽  
Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

scholarly journals A model for the free-surface flow due to a submerged source in water of infinite depth

1998 ◽  
Vol 39 (4) ◽  
pp. 528-538 ◽  
Author(s):  
J.-M. Vanden-Broeck
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Surface Condition ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Nonuniform Distribution ◽  
Infinite Depth ◽  
Collocation Points ◽  
Series Truncation Method

AbstractWe consider a free-surface flow due to a source submerged in a fluid of infinite depth. It is assumed that there is a stagnation point on the free surface just above the source. The free-surface condition is linearized around the rigid-lid solution, and the resulting equations are solved numerically by a series truncation method with a nonuniform distribution of collocation points. Solutions are presented for various values of the Froude number. It is shown that for sufficiently large values of the Froude number, there is a train of waves on the free surface. The wavelength of these waves decreases as the distance from the source increases.

Boundary Layer Measurements at an Internal Free Surface in a Partially Filled Horizontal and Rapidly-Rotating Container

1989 ◽  
Vol 111 (4) ◽  
pp. 457-463 ◽  
Author(s):  
T. J. Singler
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Circular Cylinder ◽  
Steady Flow ◽  
Symmetry Axis ◽  
Azimuthal Velocity ◽  
Free Surfaces ◽  
Horizontal Circular Cylinder ◽  
Good Agreement

Steady flow in a partially filled horizontal circular cylinder rotating rapidly about its symmetry axis is investigated experimentally. Radial boundary layer profiles of the azimuthal velocity in the neighborhood of the internal free surface are reported for a range of inverse Froude numbers and for two types of free surfaces. Results indicate good agreement with an existing theory.

scholarly journals A note on withdrawal through a point sink in fluid of finite depth

1996 ◽  
Vol 37 (3) ◽  
pp. 406-416 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Graeme C. Hocking ◽  
Graeme A. Chandler
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Finite Depth ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Flow Through

AbstractWithdrawal flow through a point sink on the bottom of a fluid of finite depth is considered. The fluid is at rest at infinity, and a stagnation point is present at the free surface, directly above the point sink. Numerical solutions are computed by means of the method of fundamental solutions, and it is observed that flows of this type are apparently possible for Froude number less than about 1.5. Relationships to previous work are discussed.

Opposing mixed convection flow in a wall jet over a horizontal plate

1997 ◽  
Vol 342 ◽  
pp. 355-375 ◽  
Author(s):  
F. J. HIGUERA
Keyword(s):  
Froude Number ◽  
Numerical Solutions ◽  
Separation Bubble ◽  
Adverse Pressure Gradient ◽  
Asymptotic Limit ◽  
Convection Flow ◽  
Separate Analysis ◽  
Wall Jet ◽  
Horizontal Plate ◽  
Recirculation Bubble

The coupling of the temperature and velocity fields by buoyancy in a laminar two-dimensional wall jet over a finite-length horizontal plate is studied numerically and analytically in the asymptotic limit of infinite Reynolds number. Two configurations are considered leading to a cold layer of fluid over the plate, namely an ambient-temperature jet over a cooled plate and a cold jet over an insulated plate. In both cases buoyancy generates an adverse pressure gradient that may separate the flow if the Froude number is sufficiently small and always makes the solution everywhere over the plate dependent on the conditions at the downstream boundary. In the limit of very small Froude number separation occurs in a viscous–inviscid interaction region near the origin of the jet, leading to a separation bubble that covers a fraction of the plate dependent on the Prandtl number. The scalings of the solution in this asymptotic limit are obtained by order of magnitude estimations in the different regions of the bubble and in the buoyancy-dominated flow beyond the bubble, and the results are checked against the numerical solutions of the boundary layer equations. A separate analysis is carried out for very large Prandtl numbers showing that the recirculation bubble is then much shorter than the plate, also in agreement with the numerical results.

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