Topics in Shear Flow Chapter 9 – The Plane Jet Donald Coles! Professor of Aeronautics, Emeritus California Institute of Technology Pasadena, California Assembled and Edited by !Kenneth S. Coles and Betsy Coles Copyright 2017 by the heirs of Donald Coles Except for figures noted as published elsewhere, this work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License DOI: 10.7907/Z90P0X7D The current maintainers of this work are Kenneth S. Coles ([email protected]) and Betsy Coles ([email protected]) Version 1.0 December 2017 Chapter 9 THE PLANE JET Consider a plane jet issuing into a stagnant fluid from a slit in a plane wall, as shown schematically in FIGURE 9.1. This configuration is often used in experimental work for the sake of its standard geometry. The most important quantity in any description of such a jet flow, whether laminar or turbulent, is J, the initial flux of momentum per unit span. As the flow develops in the downstream direction, this ini- tial momentum is conserved as it is gradually transferred from the jet fluid to the ambient fluid by shearing stresses. The rate of momentum transfer will depend not only on the nature of these stresses, but also on the relative densities of the two fluids, if these are different, and on the effect of real rather than ideal initial and boundary conditions. In any case, the total rate of fluid flow in the jet will increase contin- uously in the downstream direction as external fluid is entrained. It is this entrainment process that dominates most practical problems. For laminar flow, the solution of the boundary-layer problem is known in closed form, and the absence of a dimensionless parameter implies that all laminar plane jets are equivalent. So are all turbulent jets, for the same reason. However, for turbulent flow the growth rate is relatively rapid, and a boundary-layer approximation may not be appropriate. A simple generalization of the classical flow is obtained if the wall is absent and the jet issues from a line momentum source into 407 408 CHAPTER 9. THE PLANE JET Figure 9.1: A schematic representation of the laminar plane jet out of a vertical wall. J is the rate of momentum flux per unit span. a parallel moving stream. A small enough velocity difference, with or without a pressure gradient, may allow the jet problem to be linearized, thus providing a link with the problem of the plane wake. Even if the velocity difference is not small near the origin, it is likely to become small farther downstream, unless the pressure gradient is specially tailored to maintain a state of overall similarity. In some practical applications, the jet may issue into a confined channel and act primarily as a jet pump or as a device for thrust augmentation. In other geometries, the objective may be to shield a region by means of a jet curtain, or to modify the flow around a lifting surface by means of a jet flap, or to exploit the Coanda effect, which is the tendency for jet entrainment to evacuate an unvented region on one side or the other to a point where the jet sheet must curve toward the unvented region in order to realize the required pressure gradient normal to the local flow direction. Finally, multiple jets are known to interact strongly with each other under certain conditions. The flow immediately downstream of 9.1. LAMINAR PLANE JET INTO FLUID AT REST 409 a monoplane grid or cascade, for example, might be considered as an array of plane wakes or as an array of plane jets, depending on the solidity of the grid. In the latter case, large-scale instabilities may occur if entrainment by a jet tends to favor one unvented region over another, or if jets compete with each other in entraining fluid from the same unvented or partially vented region. FIGURE 9.2 shows a striking example of the second circumstance. 9.1 Laminar plane jet into fluid at rest 9.1.1 The equations of motion The classical laminar plane jet in an incompressible fluid is an unex- ceptional flow in terms of similarity arguments. Appropriate velocity components are (u; v) in rectangular coordinates (x; y). The mo- tion is two-dimensional in the xy-plane and is symmetrical about the plane y = 0. The pressure and density are constant everywhere. The laminar boundary-layer equations are @u @v + = 0 ; (9.1) @x @y @u @u @2u ρ u + v = µ : (9.2) @x @y @y2 Boundary conditions that require the jet flow to be symmetric about the plane y = 0 and to vanish for large y are (x; 0) = 0 or v (x; 0) = 0 ; (9.3) u (x; ±∞) = 0 ; (9.4) where is a stream function that links the velocities u and v through the relations @ @ u = ; v = − ; (9.5) @y @x and thus satisfies the continuity equation (9.1) identically. Because pressure forces are neglected in the boundary-layer approximation, the momentum flux in the body of the jet must be 410 CHAPTER 9. THE PLANE JET Figure 9.2: Visualization, using smoke filaments, of flow through a monoplane grid of cylindrical rods. The blockage is large enough to excite an entrainment instability (figure 3, plate 1, of Bradshaw 1965). Photograph courtesy of Cambridge University Press. 9.1. LAMINAR PLANE JET INTO FLUID AT REST 411 conserved during the mixing. If the continuity equation (9.1) is mul- tiplied by ρu and added to the momentum equation (9.2), the result is @uu @uv @2u ρ + = µ : (9.6) @x @y @y2 Formal integration with respect to y and use of the boundary condi- tions yields 1 d Z ρ uu dy = 0 (9.7) dx −∞ or 1 Z ρ uu dy = constant = J: (9.8) −∞ The last expression is one form of the momentum-integral equa- tion, which plays a role in every kind of boundary-layer problem. Equation (9.8) is derived from equation (9.2), but contains new in- formation because it generates the conserved quantity J as an im- portant dimensional parameter. Equation (9.8) also incorporates the boundary conditions at infinity, a point that will be developed later. Physically, the integral J represents the flux of momentum per unit time per unit span, or equally the reaction force of the jet at the origin per unit span. I have deliberately written the integrand as uu rather than u2 because the two velocities have different physical meanings. One is momentum per unit mass, and the other is volume flux per unit area per unit time. 9.1.2 Dimensional properties Intrinsic scales. If geometric details near the origin are ignored, the important physical parameters for the laminar plane jet are J, ρ, and µ. In terms of fundamental units M (mass), L (length), and T (time), denoted here and elsewhere by boldface symbols, these parameters have the dimensions M M M [J ] = ; [ρ] = ; [µ] = ; (9.9) T2 L3 LT 412 CHAPTER 9. THE PLANE JET where \ [:::] = " means \the dimensions of ::: are," and where force is replaced for dimensional purposes by mass times acceleration. Let these statements now be interpreted as defining equations for intrin- sic scales M, L, and T. That is, write M M M ≡ J; ≡ ρ ; ≡ µ : (9.10) T2 L3 LT These three definitions form an algebraic system that can be solved uniquely (in this particular instance) for the three quantities M, L, and T; ρ4ν6 ρν2 ρ2ν3 M = ; L = ; T = ; (9.11) J 3 J J 2 where ν = µ/ρ, and for the derived quantity U; L J U = = : (9.12) T ρν That the parameters µ and J should appear only in the kinematic combinations µ/ρ = ν and J/ρ is implied by the form of equations (9.2) and (9.8), respectively. Note from equations (9.11) and (9.12) that the intrinsic length and velocity scales L and U have small mag- nitudes, in the sense that their product corresponds to unit Reynolds number; UL = 1 : (9.13) ν Equations (9.11) define dimensional scales that can be used to make the problem dimensionless at the outset, without regard to the question of similarity. Such an exercise serves little purpose except to confirm, as one consequence of a dimensional inspection at the lowest possible level, that there is no dimensionless combination of J, ρ, and ν that can differ from one experiment to another. (cf. an appeal to the Buckingham Π theorem. How is round jet different?). In other words, there is only one laminar plane jet. Local scales. Some standard variations on the theme of di- mensions now follow. Ignore temporarily the intrinsic scales L and U just defined, and suppose instead that the flow has local length and 9.1. LAMINAR PLANE JET INTO FLUID AT REST 413 velocity scales L(x) and U(x), whose nature has to be determined. Similarity implies that the streamwise velocity should have the form u y = g : (9.14) U L The form of the stream function follows on integration of the first of equations (9.5); y 1 Z y = g dy = f = f(η) : (9.15) UL L L 0 Because L(x) and U(x) are designed to represent the local physical extent and the local velocity magnitude in the jet flow, the func- tion f and its derivatives will be of order unity. Substitution in the momentum equation (9.2) gives eventually L dUL L2 dU f 000 + ff 00 − f 0f 0 = 0 ; (9.16) ν dx ν dx where primes indicate differentiation with respect to y=L = η.