The maximum angle through which a sonic m 1 flow can be turned around a convex corner is calculated for m for an ideal gas, it is expressed as follows. For example, it was demonstrated that, if the coldspray nozzle is designed in such a way that at. Prandtlmeyer expansion waves when a supersonic flow is turned around a corner, an expansion fan occurs producing a higher speed, lower pressure, etc. Find the mach number and pressure of the flow downstream of the corner. M1mach number of stream flow m2mach number of flow after passing through normal shock wave nuprandtlmeyer angle, deg, for expanding flow. Mod01 lec40 introduction to expansion wave and prandtl meyer flow. When possible, these functions assume perfect gas environments. The quantitative problem of prandtlmeyer expansion wave consists in determination of. The real counterpart of the prandtlmeyer expansion is the isentropic compression of supersonic flow, which can be envisaged, for example, as the inverse flow in a laval nozzle. Prandtlmeyer flow article about prandtlmeyer flow by. Modeling liquid propellant rocket nozzle using matlab and. Pdf numerical formulation for flow computation in 2d supersonic. The laval nozzle expansion between nozzle throat and contour inflection point, fig. Because it is invalid to apply the transformation at these speeds, the predicted singularity.
Examples functions release notes pdf documentation. The figure shows an example of a simple wave lying. Then the downstream pressure and temperature are determined from the isentropic flow relations note that this is an expansion, and mach number increases while pressure and. Quick reference to sdt functions and demonstation programs pdf cantera format. If the preliminary research aerodynamic design to land on mars, or prandtlm aircraft, is successful, it could collect and transmit valuable information back to earth. Direct mach number to angle and inverse angle to mach number tables are given on page 91, for a range of mach number from i to 5, for intervals of 0.
Mod01 lec23 oblique shock waves prandtl meyer waves. Calculate prandtlmeyer functions for expansion waves matlab. The amount of the increase depends on the incoming mach number and the angle of the expansion. Prandtls presentation was only 10 minutes long, but that was all the time needed to describe a new concept that would revolutionize the understanding and analysis of fluid dynamics. One of the presenters at the congress was ludwig prandtl, a 29yearold professor at the technische hochschule equivalent to a us technical university in hanover. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at. Calculates prandtlmeyer function and polar expanded from cj state. Concluding example, shock diamonds and shockexpansion. Mod01 lec40 introduction to expansion wave and prandtl. Matlab code for the numerical solution of a prandtlmeyer.
The value of the prandtlmeyer function is therefore called the prandtlmeyer angle. Now analyzing the flow we see that l1 l2 for flow on convex corner only if n2 is greater than n1. The physical interpretation of the prandtlmeyer function is that it is the angle through which you must expand a sonic m1 flow to obtain a given mach number. Air flowing with a velocity of 680ms m2 was simulated over a wedge to study the expansion. Shocks are created when an object moves at a speed greater than that of sound in that medium. In prandtlmeyer angle input mode, mach outputs are the same size as the array input or array inputs. Ludwig prandtl and his student theodor meyer first worked out a theory for centered expansion waves in 19071908, and hence such waves are commonly denoted as prandtlmeyer expansion waves. A similar accuracy of the approximate inverse relations over the finite difference solution may exist provided that the step size in a particular problem is sufficiently large. This function solves for the flow and wave geometry in a simple wave region terminated by a characteristic of. But n2and n1 are related through normal shock wave relations. Prandtlmeyer tables staff posted on october 20, 2006 prandtlmeyer tables. The prandtlglauert singularity is a theoretical construct in flow physics, often incorrectly used to explain vapor cones in transonic flows. Prandtlmeyer angle and mach angle for expanding flows.
This program solves the prandtlmeyer expansion wave problem by using the maccormacks predictorcorrector technique, as described in the book computational fluid dynamics. Shock are sudden change in the physical properties read more. It is the prediction by the prandtlglauert transformation that infinite pressures would be experienced by an aircraft as it approaches the speed of sound. Just as we saw with shock waves, if we apply conservation of mass and momentum across a single wave, the tangential velocity is unchanged. Hall published a paper in the aeronautical journal in which he described an rational polynomial function that approximated the inverse prandtlmeyer function with accuracy sucient for most engineering purposes.
Envisioned with a wingspan of two feet and weighing less than three pounds, the aircraft would be able to. The basic working environment used for flow computation is matlab. For a given prandtlmeyer expansion, the upstream mach number is 3 and the pressure ratio across the wave is 0. Calculate flow relations for fanno lines, isentropic, normal shock, prandtlmeyer functions, and rayleigh lines. Example of the variation in aeroelastic system parameters produced by. The prandtlmeyer equation expresses the relation between the turning angle and the upstream and downstream mach numbers.
Prandtl meyer 18 ae3450 school of aerospace engineering copyright 2001 by jerry m. Calculate the angles of the forward and rearward mach lines of the expansion fan. Calculate prandtlmeyer functions for expansion waves. Isentropic expansion calculations this form calculates properties of a supersonic flow through an expansion angle. Flow on convex corner to determine the type of waves occur at the convex corner in supersonic flow, let us assume that these waves are oblique shock waves. In aerodynamics, the prandtlmeyer function describes the angle through which a flow turns isentropically from sonic velocity m1 to a mach m number greater than 1.
Chenzhangzhu 8, but also l1stable with respect to steady small perturbation of both the ramp slope and the incoming steady upstream. For the unbounded corner expansion, the wave structure forms a twodimensional distribution. Pdf this project work report provides a full solution of the governing euler equations in strong conservation form for the prandtlmeyer expansion. Prandtlmeyer angle data isentropic point expansion data normal shock wave data mach angle data oblique shock wave data. The matlab functions described here provide the means to solve for or plot the flow in.
The fan consists of an infinite number of mach waves, diverging from a sharp corner. Prandtlm work on the prandtld led to a concept for a mars airplane. If youve ever taken a gas dynamics compressible flow course, youve probably spent hours flipping through the tables at the back of the book to find the correct prandtlmeyer pm angle for your mach number, or vice versa. Prandtl meyer expansion wave solver in matlab download. An obvious advantage of the inverse relations is the convenience of their employment. Definition of nomenclature for prandtlmeyer expansion calculations. Then the downstream prandtlmeyer function becomes now ma 2 is found from the prandtlmeyer relation, which is now implicit.
Pdf this report deals with the computation of flow in the 2d supersonic nozzle. For example, in the channel formed by any two streamlines, the forward flow is expansive. In prandtlmeyer flow and the method of characteristics, calculate the important. Prandtl meyer expansion fan problem given upstream conditions 1 and turning angle. Functions in the toolbox can be used together to solve more complex compressible flow.
A supersonic expansion fan, technically known as prandtlmeyer expansion fan, a twodimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The solution technique is to consider that the flow originated at mach 1 where the prandtlmeyer angle. The prandtlmeyer angle is the angle change required for a. Python and matlab libraries and demonstration programs can be downloaded as zip. Pressure, mach number and temperature changes through normal shock waves. Previously, we examined 1 supersonic flow over sharp. Matlab calculate flow conditions at the throat and nozzle exit use prandtlmeyer expansion function to calculate left and right running characteristic lines create 2d plot of calculated nozzle shape export nozzle wall points into an excel file import wall coordinates into solidworks and generate 3d model 2d nozzle plot 3d nozzle model nozzle. Find materials for this course in the pages linked along the left. The required input is the mach number of the upstream flow and the expansion angle theta. The user should note that the scope of this guide is limited to documenting the individual functions and providing instruction in using them to solve simple compressible flow examples.
For example, in the channel formed by any two streamlines, the forward. Variation in flow properties depend on the interaction of expansion waves that occur throughout. The centered wave, more often called a prandtlmeyer expansion fan, is the. A detailed derivation of the method is outside the scope of this example analysis. Each function has an mtype argument that lets you specify the inputs for the flow. Expansion waves prandtl meyer flow shock wave mach. The following matlab project contains the source code and matlab examples used for prandtl meyer expansion wave solver. The matlab functions described here provide the means to solve for or plot the. In this project the prandtlmeyer expansion fan was simulated to study compressible flow.
A twodimensional, supersonic fluid flow in which an initially uniform flow passes a sharp, convex corner in a boundary, resulting in expansion of the fluid explanation of prandtlmeyer flow. Aerospace toolbox provides tools and functions for analyzing the navigation and environment of aerospace vehicles and visualizing their flight using standard cockpit instruments or a flight simulator. User guide for compressible flow toolbox ntrs nasa. Numerical investigation of homogeneous condensation in. How do you compute the inverse of the prandtl meyer function. Multiple shock waves are observed in the numerical schlieren image, which are caused by a sequence of quenching by the condensation shock and rebuilding by the prandtlmeyer expansion of the condensation.
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