Concept explainers
Assuming the velocity field given in Problem 2.6 pertains to an incompressible flow, calculate the stream function and velocity potential. Using your results, show that lines of constant
To find:
The stream function and velocity potential for the given velocity functions and to prove it’s perpendicularity.
Answer to Problem 2.11P
Equation of the velocity potential is
Equation of the stream function is
Explanation of Solution
Given:
The horizontal and vertical velocity component of velocity is given as below ( problem 2.6).
Stream function and velocity potential has to be found and prove perpendicularity of the stream function and velocity potential.
Stream function:
The equation of the stream function is given as follows:
Equation of the stream function is
Velocity potential:
The equation of the velocity potential is given as follows:
Equation of the stream function is
Proof of the perpendicularity:
Differentiate equation (2) and (3) with respect to x keeping φ and ψ constant.
Similarly,
Compare equations 7 and 9.
Hence,the stream function and velocity potential are perpendicular to each other.
Want to see more full solutions like this?
Chapter 2 Solutions
ENG ME 421 AERODYNAMICS >CUSTOM<
- in the X-Y plane; permanent, 2B, in an uncompressed flow, a, b and C are constant there is a velocity field defined as follows. a-) Continuity equation show that it is provided. b-) calculate the pressure as a function of x and y?arrow_forwardConsider a velocity field where the x and y components of velocity aregiven by u = cx and v = −cy, where c is a constant. Assuming the velocity field given is pertains to an incompressible flow, calculate the stream function and velocity potential.Using your results, show that lines of constant φ are perpendicular to linesof constant ψ.arrow_forwardThe velocity components of an incompressible, two-dimensional velocity field are given by the equations Show that the flow satisfies continuity. (b) Determine the corresponding stream function for this flow field. (c) Determine if the flow is irrotational.arrow_forward
- Consider steady flow of water through an axisymmetric garden hose nozzle. Along the centerline of the nozzle, the water speed increases from uentrance to uexit as sketched. Measurements reveal that the centerline water speed increases parabolically through the nozzle, calculate the fluid acceleration along the nozzle centerline as a function of x and the given parameters.arrow_forwardA velocity profile for water is given as a function of x, y and z. How can we determine if the profile is a physically possible flow field? A) If the sum of the derivatives of each component with respect to their flow direction = 0. B) If the flow is irrotational. C) If the vorticity equals 0. D) If we can determine a velocity potential.arrow_forwardfor the following flows, find the equation of the streamline through(1,1). v= -y^(2)i-6xjarrow_forward
- Consider an airplane flying with a velocity of 60 m/s at a standard altitude of 3km. At a point on the wing, the airflow velocity is 70 m/s. Calculate the pressure at this point. Assume incompressible flow.arrow_forwardConsider steady, incompressible, parallel, laminar flow of a film of oil falling slowly down an infinite vertical wall. The oil film thickness is h, and gravity acts in the negative z-direction. There is no applied (forced) pressure driving the flow—the oil falls by gravity alone., except for the case in which the wall is inclined at angle ?. Generate expressions for both the pressure and velocity fields. As a check, make sure that your result agrees with that of when ? = 90°. [Hint: It is most convenient to use the (s, y, n) coordinate system with velocity components (us, ?, un), where y is into the page in Fig. Plot the dimensionless velocity profile us* versus n* for the case in which ? = 60°.]arrow_forwardA horizontal pipe of diameter 25cm has a constriction of diameter of 5cm. The velocity of the water in the pipe is 1cm/s and the pressure is 10⁵ Pa. Calculate the velocity and pressure in the constriction. Include an illustration or drawing of the situationarrow_forward
- 1. Find the stream function for a parallel flow of uniform velocity V0 making an angle α with the x-axis. 2. A certain flow field is described by the stream function ψ = xy. (a) Sketch the flow field. (b) Find the x and y velocity components at [0, 0], [1, 1], [∞, 0], and [4, 1]. (c) Find the volume flow rate per unit width flowing between the streamlines passing through points [0, 0] and [1, 1], and points [1, 2] and [5, 3].arrow_forwardConsider the velocity field given by u = y/(x2 + y2) and v = −x/(x2 + y2). Calculate the equation of the streamline passing through the point (0, 5).arrow_forwardConsider the velocity field given by u = y/(x2 + y2) and v = −x/(x2 + y2). Calculate the vorticity.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY