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.

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.

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.

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.

scholarly journals Flow Pattern Map of Flow Boiling in a Rectangular Channel Filled with Porous Media

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2440
Author(s):  
Youngwoo Kim ◽  
Dae Yeon Kim ◽  
Kyung Chun Kim
Keyword(s):  
Porous Media ◽  
Flow Pattern ◽  
Flow Regime ◽  
Annular Flow ◽  
Flow Boiling ◽  
Rectangular Channel ◽  
Flow Patterns ◽  
Flow Pattern Map ◽  
Mist Flow ◽  
Churn Flow

A flow visualization study was carried out for flow boiling in a rectangular channel filled with and without metallic random porous media. Four main flow patterns are observed as intermittent slug-churn flow, churn-annular flow, annular-mist flow, and mist flow regimes. These flow patterns are clearly classified based on the high-speed images of the channel flow. The results of the flow pattern map according to the mass flow rate were presented using saturation temperatures and the materials of porous media as variables. As the saturation temperatures increased, the annular-mist flow regime occupied a larger area than the lower saturation temperatures condition. Therefore, the churn flow regime is narrower, and the slug flow more quickly turns to annular flow with the increasing vapor quality. The pattern map is not significantly affected by the materials of porous media.

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 The hodograph transformation in trans-sonic flow - IV. Tables

1947 ◽  
Vol 192 (1028) ◽  
pp. 135-142 ◽  
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Compressible Flow ◽  
Variable Quantity ◽  
Supersonic Region ◽  
Flow Round ◽  
Sonic Flow ◽  
In Trans ◽  
Hodograph Transformation ◽  
Subsonic Region

In Part III of this series (Lighthill 1947 c ) it was shown that the problem of finding a plane steady adiabatic compressible flow round a body which reduces to a given incompressible flow (with or without circulation) when the Mach number tends to zero can be solved, in the subsonic region at least, if certain functions ψ n (ז), whose properties are set out at length in Part II, are known, with their derivatives, for positive integral n and for ז < ( γ – 1)/( γ + 1), where γ is the adiabatic index. (The functions ψ n (ז) depend on γ .) In the supersonic region the same functions may be needed for higher values of ז; but other values of n would also be needed. In Part I (Lighthill 1947 a ) it is shown how a knowledge of the ψ n (ז) may help in the design of symmetrical channels. It is also well known (see, for example, Chaplygin 1904) that the exact solution of ‘free streamline’ problems in subsonic compressible flow may be achieved by means of the functions ψ n (ז). Finally, it seems likely that future theories of compressible flow will use the functions ψ n (ז), especially for positive integral n . These considerations have led us to tabulate the functions ψ n (ז) and ψ ' n (ז) for values of ז between 0 and ½ at intervals of 0∙02 and for the values 1, 2, 3, . . . , 15 of n taking γ = 1∙4 ( T = 1/6 then corresponds to Mach number 1 and T = ½ and to Mach number √5.) We have taken γ = 1∙4 for air rather than γ = 1∙405, because, while both values have been widely adopted, the little experimental evidence relating to this rather variable quantity points to the lower value as nearer the truth; it is also considerably simpler to use.

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