Technology & Engineering
Velocity Profile
Velocity profile refers to the variation of velocity within a fluid or gas as it flows through a particular area, such as a pipe or channel. It is typically represented graphically, showing how the velocity changes across the cross-section of the flow. Understanding the velocity profile is crucial in fluid dynamics and engineering, as it impacts factors like pressure drop and flow distribution.
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eBook - ePubIntroduction to Engineering Mechanics
A Continuum Approach, Second Edition
- Jenn Stroud Rossmann, Clive L. Dym, Lori Bassman(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
18Fluid Dynamics: Governing Equations
In Chapter 17 , we considered cases in which there was no relative motion of fluid particles—no velocity gradients, and thus no shear stress. Now we will consider the somewhat more interesting flows in which velocity gradients and accelerations do appear.18.1 Description of Fluid Motion
You have probably seen the car companies’ ads featuring this year’s models in wind tunnels, with smoke tracing the flow of air over the cars’ streamlined curves. There is a mathematical way to define the equations of these smoke traces, and a physical interpretation of them, that we will find quite useful in our discussion of fluid dynamics.The velocity field specifies the instantaneous speed and direction of the motion of all points in the flow. A streamline is everywhere tangent to the velocity field and so reflects the character of the flow field. Streamlines are instantaneous, being based on the velocity field at one given time. The smoke traces mentioned above and illustrated in Figure 18.1 are streaklines, which include all the fluid particles that once passed through a certain point. We may also describe a flow field with pathlines, which represent the trajectory traced out by a given fluid particle over time. When the flow is steady, or independent of time t, the streamlines, streaklines, and pathlines coincide.For a two-dimensional flow field, we can find the equations of streamlines by applying their definition. Since streamlines are tangent to velocity, the slope of a streamline must equals the tangent of the angle that the velocity vector makes with the horizontal, as shown in Figure 18.2
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
In Figure 3.2 a—flow in a pipe—the velocity at any cross section is parabolic. The velocity is thus a function of the radial coordinate; a gradient exists. In addition, a pressure gradient exists in the axial direction that maintains the flow. That is, a pressure difference from inlet to outlet is imposed on the fluid that causes flow to occur. The flow is one-directional, but because we have both a velocity and a pressure gradient, it is two-dimensional. Another example of two-dimensional flow is given in Figure 3.2 b. At the constant-area sections, the velocity is a function of one variable. At the convergent section, velocity is a function of two space variables. In addition, a pressure gradient exists that maintains the flow. Figure 3.2 c gives another example of a one-directional, two-dimensional flow; gradients exist in two-dimensions. FIGURE 3.2 Two-dimensional flow. It is possible to assume one-dimensional flow in many cases in which the flow is two-dimensional to simplify the calculations required to obtain a solution. An average constant velocity, for example, could be used in place of a parabolic profile, although the parabolic profile gives a better description of the flow of real fluids because velocity at a boundary must be zero relative to the surface (except in certain special cases such as rarefied gas flows). A flow is said to be three-dimensional when the fluid velocity or flow parameters vary with respect to all three space variables. Gradients thus exist in three directions. Flow is said to be steady when conditions do not vary with time or when fluctuating variations are small with respect to mean flow values, and the mean flow values do not vary with time. A constant flow of water in a pipe is steady because the mean liquid conditions (such as velocity and pressure) at one location do not change with time. If flow is not steady, it is called unsteady : mean flow conditions do change with time
eBook - ePub- Keith J. Moss(Author)
- 2015(Publication Date)
- Routledge(Publisher)
L , therefore, for most practical applications becomes the straight length of the pipe or duct being considered.Figure 6.8 shows the formation of the boundary layer in laminar and turbulent flow in a straight pipe.Velocity Profile for laminar and turbulent flow in straight pipes
Due to the surface resistance at the boundary walls of the pipe and the viscosity of the fluid, maximum velocity occurs at the pipe centreline and zero velocity at the pipe wall. The velocity gradient u /L may be obtained at any point P on the Velocity Profile, Figure 6.6 .Figure 6.8 Formation of the boundary layer in a pipe.At the pipe centreline the Velocity Profile u /L = zero. Thus the boundary layer is the layer of fluid contained in a Velocity Profile up to the point where the velocity gradient is zero.Boundary layer separation
The separation of the boundary layer from the solid boundary surface does not occur in straight pipes or ducts. This is because there is a steady static pressure loss in the direction of flow. It does occur, however, in tees, Y junctions, bends and gradual enlargements, and its effects on pressure losses through fittings are analysed in Chapter 7 .It can be shown that in each of the fittings identified here, there is a momentary gain in static pressure as the fluid passes through. This is most commonly noted in the gradual enlargement in which the gain in static pressure is held. The gain in static pressure is at the expense of a corresponding loss in velocity pressure whether it is momentary or otherwise and this causes the boundary layer to separate from the solid boundary surface. It rejoins at some point downstream. Figures 6.9 –6.12
- Amithirigala Widhanelage Jayawardena(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
These profiles can be plotted on logarithmic scales with dimensionless axes: Ordinate u u * where u * = τ 0 / ρ is the shear velocity Abscissa y u * / ν The relationship between these two dimensionless variables is known as the ‘law of the wallʼ. It is a universal plot for smooth surfaces. 7.3.4.5 Universal Velocity Profile near a wall The universal Velocity Profile, obtained by integrating the Navier−Stokes equation (Yuan, 1967; pp 372–374) is given by u u * = 1 k (l n u * y ν − l n β) (7.71) where k and β are experimentally determined. With Nikuradse ’s experimental results, it takes the form u u * = 5.75 log (u * y ν) + 5.5 (7.72) In the laminar sublayer, the linear Velocity Profile is valid: u u * = u * y ν (7.73) In the transition zone, von. Karman suggested u u * = 11.5 log (u * y ν) − 3.05 (7.74) 7.4 Viscous flow between parallel plates When a viscous fluid flows between two fixed parallel plates, the resulting flow is known as Poiseuille flow. This is discussed in Section 7.2.2. The Velocity Profile is u = 1 μ ∂ p ∂ x y 2 2 − 1 μ ∂ p ∂ x h 2 2 = 1 2 μ ∂ p ∂ x (y 2 − h 2) (7.75) and the maximum velocity. is u max = u 0 = − 1 2 μ d p d x h 2 (7.76) T h e d i s c h a r g e Q = ∫ − h h b u d y = ∫ − h h b 1 2 μ d p d x (y 2 − h 2) d y = 2 h 3 b 3 μ (− d p d x) (7.77) where b is the width of the plate and the average. velocity V is given. as V = Q A = h 2 3 μ (− d p d x) (7.78) S h e a r s t r e s s τ x y = μ d u d y = μ h 2 2 μ (− d p d x) (− 2 y h 2) (7.79) W a l l s h e a r s t r e s s (τ 0) y = h = μ[-. -=PLGO-SEPARATOR=--]h (− d p d x) (7.80a) (τ 0) y = − h = − μ h (− d p d x) (7.80b) When one plate is fixed and the other one is moving at a constant velocity, the resulting flow is known as Couette flow
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