流体力学上课讲义第四章-1

Mass of element
Any pressure acting on the side of the cylindrical element will not contribute to a force in the l direction.
W Al
Force due to pressure in the l direction Force due to gravity in the l direction
For incompressn
Euler’s Equation
( p z ) al l
(4.8)
• Euler’s equation shows that • Acceleration is equal to the change in piezometric pressure with distance • The minus sign means that the acceleration is in the direction of decreasing piezometric pressure • In a static fluid, Euler’s equation reduces to
When the flow is steady, streamlines, pathlines, and streaklines are identical.
Figure 4.6 Streamlines, pathlines, and streakline for an unsteady flow field. Both the pathline and streakline provide a history of the flow field, and the streamlines indicate the current flow pattern.
u u u
u does not
One-dimensional and multi-dimensional flows
The dimensionality of a flow field is characterized by the number of spatial dimensions needed to describe the velocity field. One-dimension
V 0 t V 0 t
For example, the flow rate changes due to a valve closing or opening
• Streamline: a curve that is everywhere tangent to the instantaneous local velocity vector. • Pathline: the actual path traveled by an individual fluid particle over some time period. • Streakline: the locus of fluid particles that have passed sequentially through a prescribed point in the flow.
FIGURE 4–15 Cimbala & Cengel A pathline is formed by following the actual path of a fluid particle
FIGURE 4–16 Cimbala & Cengel Pathlines produced by white tracer particles suspended in water and captured by time-exposure photography; as waves pass horizontally, each particle moves in an elliptical path during one wave period.
Figure 4.2 Predicted streamline pattern over the Volvo ECC prototype.
Pathline
The actual path traveled by an individual fluid particle over some time period.
Figure 4.3 Fluid particle moving along a pathline.
Figure 4.4 Uniform flow in a pipe.
Nonuniform flow
Figure 4.5 Flow patterns for nonuniform flow. (a) Converging flow. (b) Vortex flow.
Figure. 4.1 Flow through an opening in a tank and over an airfoil section
Streamline
• Streamline cannot be directly observed experimentally except in steady flow fields, in which they are coincident with pathlines and streaklines
Laminar vs. Turbulent flow
Laminar flow Turbulent flow
• Since turbulent flow is unsteady, the velocity at any point fluctuates with time. • The standard approach to treating turbulent flow is to represent the velocity as a time-averaged value plus a fluctuating quantity, • A turbulent flow is often designated as “steady” if change with time.
FIGURE 4–18 Cimbala & Cengel Streaklines produced by colored fluid introduced upstream; since the flow is steady, these streaklines are the same as streamlines and pathlines.
FIGURE 4–12 Cimbala & Cengel Spinning baseball. Flow field visualized using smoke.
Streamline
• Streamline: a curve that is everywhere tangent to the local velocity vector at that instant. • The tangent of the streamline at a given point gives the direction of the velocity vector. • The streamline, however, does not indicate the magnitude of the velocity.
z z Fgravity W Al l l
Substituting the mass and the forces on the element into Eq. (4.6) yields
Dividing through by Al
Taking the limit l 0
Two-dimension
Three-dimension
4.3 Euler’s Equation
Consider a fluid element oriented and accelerating in an arbitrary l direction. Assume that the viscous forces are zero. Applying Newton’s second law in the l direction
Uniform vs. Nonuniform flow
Velocity of the fluid
Uniform flow: the velocity does not change along the a fluid path
Nonuniform flow: the velocity changes along the a fluid path
Steady vs. Unsteady flow
Steady flow : the velocity at a given point does not change with time Unsteady flow : the velocity at a given point changes with time
( p z ) 0 l
The hydrostatic differential equation
Laminar vs. Turbulent flow
Laminar flow: well-ordered state of flow in which adjacent fluid layers move smoothly with respect to each other
Turbulent flow: an unsteady flow characterized by intense cross-stream mixing.
Streakline
The line generated by a tracer fluid continuously injected into the flow field.