Gravitational Flow Discharge From a Horizontal Duct

1985 ◽  
Vol 52 (1) ◽  
pp. 167-171 ◽  
Author(s):  
P. Chan ◽  
T. Han ◽  
W. L. Chow
Keyword(s):  
Initial Pressure ◽  
Unknown Boundary ◽  
Physical Plane ◽  
Flow Discharge ◽  
Free Jet ◽  
Gravitational Flow ◽  
Total Head ◽  
Boundary Functions ◽  
Hodograph Plane ◽  
Hodograph Transformation

The problem of a potential flow discharge through a two-dimensional horizontal duct under the influence of gravitation is examined by the method of hodograph transformation. The stream function is considered and established in the hodograph plane, and the solution in the physical plane is established through additional integrations. The unknown boundary functions of the free jet must be determined as part of the solution. The initial pressure level and the discharge characteristics between the total head and the flow rate, have been established. Results are compared with those obtained previously by other method.

scholarly journals The hodograph transformation in trans-sonic flow. I. Symmetrical channels

1947 ◽  
Vol 191 (1026) ◽  
pp. 323-341 ◽  
Keyword(s):  
Equations Of Motion ◽  
Physical Plane ◽  
Adiabatic Flow ◽  
Independent Variables ◽  
Hodograph Plane ◽  
Sonic Flow ◽  
In Trans ◽  
Hodograph Transformation ◽  
Steady Plane ◽  
The Right

It seems likely that any general theory of compressible flow applicable to problems with regions both of sub- and supersonic flow (such problems have been called ‘trans-sonic’) must be based on the ‘hodograph transformation’ (due originally to Molenbroek 1890 and Chaplygin 1904). This is because in the hodograph plane, in which the independent variables are the magnitude and direction of the velocity, the equations of motion are linear; while in the physical plane they are not even approximately linear for trans-sonic problems. But the hodograph transformation presents difficulties quite apart from those of applying suitable boundary conditions. It has in fact singularities, notably near the sonic speed and the velocity at infinity. This fact considerably elaborates its use. In the present paper a study is initiated of the application of the hodograph transformation to trans-sonic problems by considering the steady plane adiabatic flow of a gas in symmetrical channels in which the velocity rises from zero at infinity on the left to a supersonic value at infinity on the right; this is a problem easier to begin on than those with a body inside the field of flow, which are known in practice to involve shock-waves and hence regions of non-adiabatic flow.

Discharge From a Vessel Through an Axisymmetric Control Valve

1987 ◽  
Vol 54 (2) ◽  
pp. 447-452 ◽  
Author(s):  
W. L. Chow ◽  
Alice A. L. Ting ◽  
P. H. Tsai
Keyword(s):  
Boundary Value Problem ◽  
Numerical Computation ◽  
Incompressible Flow ◽  
Boundary Value ◽  
Control Valve ◽  
Unusual Feature ◽  
Flow Discharge ◽  
Hodograph Plane ◽  
Hodograph Transformation

The problem of an incompressible flow discharge through an axisymmetric control valve has been examined through the method of hodograph transformation. The boundary value problem in the hodograph plane has an unusual feature in that part of the boundary is overlapped. Certain manipulations have been applied to produce the realistic free streamline configuration. The rate of discharge is obtained for various locations of the valve. This investigation provides the evidence that the method of hodograph transformation coupled with numerical computation is indeed effective in dealing with problems of this kind.

The Interaction Between a Jet and a Flat Plate—An Inviscid Analysis

1995 ◽  
Vol 117 (4) ◽  
pp. 623-627 ◽  
Author(s):  
W. L. Chow ◽  
Z. P. Ke ◽  
J. Q. Lu
Keyword(s):  
Flat Plate ◽  
Flow Pattern ◽  
Flow Regime ◽  
Compressible Flow ◽  
Physical Plane ◽  
Numerical Computations ◽  
Practical Applications ◽  
Hodograph Plane ◽  
Hodograph Transformation

The problem of jet-plate interaction has been examined. It is shown that the problem of this type is governed by the mechanisms of inviscid interaction. The method of hodograph transformation has been employed to formulate the problem, and the solution is obtained from numerical computations in the hodograph plane. The flow pattern in the physical plane is produced from additional integrations. Extensions to the compressible flow regime with practical applications have also been mentioned.

scholarly journals The hodograph transformation in trans-sonic flow. III. Flow round a body

1947 ◽  
Vol 191 (1026) ◽  
pp. 352-369 ◽  
Keyword(s):  
Mach Number ◽  
Incompressible Flow ◽  
Equations Of Motion ◽  
The Body ◽  
Physical Plane ◽  
Hodograph Plane ◽  
In Series ◽  
Flow Round ◽  
Sonic Flow ◽  
Hodograph Transformation

As was remarked in part I, §1, the hodograph transformation offers one of the most hopeful approaches to trans-sonic flow problems. But it is ill adapted for solving exactly the problem of flow past a contour of given shape. Boundary conditions more remote from this ideal one are necessary, chosen to give, past a contour approximating to the one desired, a flow exactly solving the equations of motion. Thus in part I (Symmetrical Channels) the velocity distribution along the axis was stipulated. In the problem of flow round a body, uniform and subsonic at infinity but possibly supersonic in certain regions, it is convenient to construct a flow such as will reduce to the incompressible flow round a body of approximately the same shape when the Mach number tends to zero. Some previous writers have sought to do this by expanding in series in different parts of the incompressible hodograph plane (at least four distinct expansions being necessary to cover the plane) and then modifying each series to allow for compressibility. While each modified series satisfied the equations of motion, they were not analytic continua­tions of each other, so their combination corresponded to no physical possibility. These statements on previous writers’ work are proved in the Appendix. In §2 a solution valid over the whole subsonic region of the physical plane is given. This solution is given in terms of integrals in the physical plane for the incompressible flow and can therefore be used when only data of the most numerical kind are available concerning this flow (to which the solution reduces when the Mach number tends to zero). In § 4 it is shown how, when an analytic series (of a very general type) is available in the incompressible flow, the solution can be continued into the super­sonic region. The solution contains an arbitrary function: so the different possible determinations of this function lead to an infinity of solutions of the compressible flow problem, all tending to the given incompressible flow as the Mach number tends to zero. It is shown that when circulation is absent all these solutions give a possible physical picture: the natural consequence is to take the simplest one, which particular solution is discussed in § 5. In § 6 it is seen that when circulation is present, however, all the solutions but one give a physical plane which does not close up behind the body. The single solution which gives a physically sensible result in this case is determined and its properties are investigated in §6. The results of part II on the fundamental functions ψ n (ז) are used throughout the work.

Discharge of a Compressible Fluid Through a Control Valve

1987 ◽  
Vol 54 (4) ◽  
pp. 955-960 ◽  
Author(s):  
Z. M. Weng ◽  
Alice A. L. Ting ◽  
W. L. Chow
Keyword(s):  
Compressible Fluid ◽  
Control Valve ◽  
Supercritical Pressure ◽  
Special Treatment ◽  
Ample Evidence ◽  
Flow Discharge ◽  
Free Jet ◽  
Sonic Line ◽  
Original Analysis ◽  
Hodograph Transformation

The original analysis on incompressible flow discharge from a vessel through an axisymmetric control valve has been extended to the discharge of a compressible fluid. The inviscid anlaysis is based on the method of hodograph transformation. While it is simple to account for the effect of compressibility for conditions of subcritical pressure ratios, special treatment must be applied to establish the sonic line and the free jet boundaries under conditions of supercritical pressure ratios. Discharge characteristics have been established for different pressure ratios and positions of the control valve. This series of investigations provides ample evidence that the hodograph transformation coupled with numerical computations is effective in dealing with problems of this nature.

Jet-Plate Interaction for Wedge-Shaped Plates of Arbitrary Angles

1997 ◽  
Vol 119 (4) ◽  
pp. 929-933 ◽  
Author(s):  
S. S. Chu ◽  
W. L. Chow
Keyword(s):  
Jet Flows ◽  
Test Cases ◽  
Rectangular Grid ◽  
Physical Plane ◽  
Flow Properties ◽  
Flow Parameters ◽  
Hodograph Plane ◽  
Free Streamlines ◽  
Hodograph Transformation ◽  
Direct Numerical Integration

An investigation has been undertaken to study the problems of jet-plate interaction through the method of hodograph transformation. The physical flow field is first transformed to a hodograph domain. By using properly selected flow parameters, the solution is established through numerical computations with rectangular grid in the hodograph plane. The resulting plate configuration, the free streamlines, and the flow properties in the physical plane are subsequently obtained through direct numerical integration. Jet flows toward wedge-shaped plates of arbitrary angles are solved to demonstrate the ability of the method. To verify the solutions, momentum principle has been employed in the physical plane for all test cases. It is found that the results obtained through this method are satisfactory.

Polygonal Approximation Method in the Hodograph Plane

1949 ◽  
Vol 16 (2) ◽  
pp. 123-133
Author(s):  
H. Poritsky
Keyword(s):  
Fluid Flow ◽  
Approximation Method ◽  
Applied Mechanics ◽  
Polygonal Approximation ◽  
Physical Plane ◽  
Flow Equations ◽  
Hodograph Plane ◽  
Superposition Of Solutions ◽  
Sixth International

Abstract This paper extends the discussion of the approximate method of integrating the equations of compressible fluid flow in the hodograph plane first presented by the author before the Sixth International Congress of Applied Mechanics, Paris, France, September, 1948. As an introduction to the discussion of the polygonal approximation method, fundamental fluid-flow equations are reviewed briefly. Determination of the flow function ψ by the “Method of Reflections” is described and an application of the method illustrated. How flow in the physical plane can be determined by superposition of solutions discussed is shown for the simpler incompressible case.

scholarly journals Steady compressible vortex flows: the hollow-core vortex array

1995 ◽  
Vol 301 ◽  
pp. 1-17 ◽  
Author(s):  
K. Ardalan ◽  
D. I. Meiron ◽  
D. I. Pullin
Keyword(s):  
Mach Number ◽  
Vortex Core ◽  
Linear Problem ◽  
Constant Pressure ◽  
Speed Ratio ◽  
Hollow Core ◽  
Physical Plane ◽  
Single Row ◽  
Transonic Shock ◽  
Hodograph Plane

We examine the effects of compressiblity on the structure of a single row of hollowcore, constant-pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters M∞ = q∞/c∞, and the speed ratio, a = q∞/qv, where qv is the speed on the vortex boundary, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, M∞, is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines.

scholarly journals Design of A Streamwise-Lateral Ski-Jump Flow Discharge Spillway

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1585 ◽  
Author(s):  
Jun Deng ◽  
Wangru Wei ◽  
Zhong Tian ◽  
Faxing Zhang
Keyword(s):  
Side Wall ◽  
Hydraulic Engineering ◽  
Limited Area ◽  
Lateral Direction ◽  
Flow Discharge ◽  
Free Jet ◽  
Plunge Pool ◽  
New Type ◽  
Solid Walls ◽  
Large Discharge

Spillway outlet design is a major issue in hydraulic engineering with high head and large discharge conditions. A new type of design for a streamwise-lateral spillway is proposed for ski-jump flow discharge and energy dissipation in hydraulic engineering. The water in the spillway outlet is constrained by three solid walls with an inclined floor, a horizontal floor on the bottom and a deflected side wall in the lateral direction. The water flow releases in a lateral direction into the plunge pool along the streamwise direction. It generates a free jet in the shape of “∩” in a limited area, causing the water to fully diffuse and stretch in the air simultaneously, and drop into the plunge pool to avoid excessive impact in the plunge pool. The formation mechanism for the flow pattern is analyzed, and the results show that the optimum inclination is an angle range of 30°~45° for a good performance of free ski-jump jet diffusion shape.

The Numerical Solution of Choked and Supercritical Ideal Gas Flow through Orifices and Convergent Conical Nozzles

1979 ◽  
Vol 21 (3) ◽  
pp. 197-203 ◽  
Author(s):  
G. M. Alder
Keyword(s):  
Numerical Solution ◽  
Gas Flow ◽  
Ideal Gas ◽  
Physical Plane ◽  
Truncation Errors ◽  
Hodograph Plane ◽  
Supersonic Region ◽  
Flow Through ◽  
Supercritical Flows ◽  
Subsonic Region

The paper describes the numerical solution of the equations of compressible flow through axisymmetric convergent nozzles. The class of supercritical flows is considered, in which the gas velocities in the jet downstream from the throat are supersonic. The subsonic region of the flowfield is solved in the hodograph plane by a finite-difference method. The supersonic region is solved in the physical plane by the method of characteristics. The stream function distribution on the sonic line is adjusted iteratively to match the boundary conditions at the lip and free streamline. Discharge coefficients are evaluated and truncation errors in the results are considered.

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