Concept explainers
In an oscillating compressible flow field the volumetric strain rate is not zero, but varies with time following a fluid particle. In Cartesian coordinates we express this as
Suppose the characteristic speed and characteristic length for a given flow field arc V and L, respectively. Also suppose that/is a characteristic frequency of the oscillation (Fig. P7-31). Define the following dimensionless variables,
Noudimensionalize the equation and identify any established (named) dimensionless parameters that may appear.
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Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
- Given the following steady, two-dimensional velocity field. [Diberi medan halaju yang mantap dan dua dimensi.] V = (u, v) = (6x + 4y – 2)i + (-6y + 4x – 5)j %3D (i) Is this flow field an incompressible flow? Prove your answer. [Adakah medan halaju ini adalah aliran tidak boleh mampat? Buktikan jawapan anda.] (ii) Is this flow field irrotational? Prove your answer. [Adakah medan halaju ini adalah aliran tidak berputar? Buktikan jawapan anda.] (iii) Generate an expression for the velocity potential function if applicable. [Terbitkan satu ungkapan bagi fungsi keupayaan halaju jika boleh dilakukan.]arrow_forwardA Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…arrow_forwardConverging duct flow is modeled by the steady, two- dimensional velocity field V = (u, v) = (U₁ + bx) i-by. For the case in which Ug = 3.56 ft/s and b = 7.66 s¯¹, plot several streamlines from x = 0 ft to 5 ft and y=-2 ft to 2 ft. Be sure to show the direction of the streamlines. (Please upload you response/solution using the controls provided below.)arrow_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_forwardA velocity field of the two-dimensional, time-dependent fluid flow is given by where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any time t > 0 and any pont x > 0, y > 0. a) Incompressibility a) Is this flow incompressible (i.e. it has zero divergence)? Yes No ди Ət b) Time derivative of flow field Calculate the time derivative of the velocity. Represent your answer in the form i+ || 3 3 u(t, x, y) =r? (x² + y² ) i− {etxtyj X уј 3 a = c) Material derivative and acceleration Calculate the material derivative of the velocity and hence the acceleration a. Represent your answer in the form Du Dt || j i+ jarrow_forwardUse Eq. dx/u =dy/v=dz/w=dr/V to find and sketch the streamlines of the followingfl ow field:u = Kx; v = -Ky; w = 0, where K is a constant.arrow_forward
- An incompressible velocity field is given by u=a(x°y²-y), v unknown, w=bxyz where a and b are constants. (a)What is the form of the velocity component for that the flow conserves mass? (b) Write Navier- Stokes's equation in 2-dimensional space with x-y coordinate system.arrow_forwardVerify whether or not the following difference representation for the continuity equation for a 2-D steady incompressible flow has the conservation property: (Ui+1,j + U₁+1, j-1 — Ui, j — Ui,j-1) (Vi+¹, j — Vi+1,j-1). Ay + 2Ax where u and v are the x and y components of velocity, respectively.arrow_forward1. Stagnation Points A steady incompressible three dimensional velocity field is given by: V = (2 – 3x + x²) î + (y² – 8y + 5)j + (5z² + 20z + 32)k Where the x-, y- and z- coordinates are in [m] and the magnitude of velocity is in [m/s]. a) Determine coordinates of possible stagnation points in the flow. b) Specify a region in the velocity flied containing at least one stagnation point. c) Find the magnitude and direction of the local velocity field at 4- different points that located at equal- distance from your specified stagnation point.arrow_forward
- CX a) Investigate whether the function o represents the velocity potential x²+y2 of a particular incompressible 2D flow, and if so, what should be the dimension of constant C which has value of 2.arrow_forward(a) Given the following steady, two-dimensional velocity field. [Diberi medan halaju yang mantap dan dua dimensi.] V = (u, v) = (8x + 6)ï + (-8y – 4)j (i) Is this flow field an incompressible flow? Prove your answer. (ii) Is this flow field irrotational? Prove your answer. (iii) Generate an expression for the velocity potential function if applicable.arrow_forwardQI A/ The inviscid, steady, and incompressible 2D flows are given by (a) o =x- 3xy (b) y = x-2xy-y? In each case, find the components of velocity in x- and y-directions.arrow_forward
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