Chapter 4 Pressure differences are (often) the forces that move fluids FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics, Spring Term 2011 e.g., pressure is low at the center of a hurricane. For your culture : The force balance for hurricanes is the pressure force vs. the Coriolis force. Thatʼs why hurricanes on the northern hemisphere always spin counter-clockwise. In a tornado, the balance is between pressure force and centrifugal acceleration; a tornado can spin either way Lagrangian and Eulerian Descriptions of Fluid Motion Eulerian: Observer stays at a fixed point in space. Lagrangian: Observer moves along with a given fluid particle. For fluid mechanics, the more convenient description is usually the Eulerian one: We consider first the Lagrangian case. The position of a fluid particle at a given time t can be written as a Cartesian vector For a complete description of a flow, we need to know at every point. Usually one references each particle to its initial position:
For your Culture : In solid mechanics, we often use Lagrangian methods. We donʼt need to follow every particle, but just some small volumes. Streamlines and Flow Patterns Figure 4.3 (p. 84) Streamlines are used for visualizing the flow. Several streamlines make up a flow pattern. When displacements get large (e.g. in fluid flow), the deforming grid gets problematic. But we sometimes use mixed approaches, e.g. Lagrangian tracers in a Eulerian frame. A streamline is a line drawn through the flow field such that the flow vector is tangent to it at every point at a given instant in time. Uniform vs. Non-Uniform Flow Using s as the spatial variable along the path (i.e., along a streamline): Examples of non-uniform flow: Flow is uniform if Examples of uniform flow: a) Converging flow: speed increases along each streamline. Note that the velocity along different streamlines need not be the same! (in these cases it probably isnʼt). b) Vortex flow: Speed is constant along each streamline, but the direction of the velocity vector changes.
Steady vs. Unsteady Flow Turbulent flow in a jet For steady flow, the velocity at a point or along a streamline does not change with time: Any of the previous examples can be steady or unsteady, depending on whether or not the flow is accelerating: Turbulence is associated with intense mixing and unsteady flow. Flow around an airfoil: Partly laminar, i.e., flowing past the object in layers (laminae). Turbulence forms mostly downstream from the airfoil. Flow inside a pipe: Laminar Turbulent (Flow becomes more turbulent with increased angle of attack.) Turbulent flow is nearly constant across a pipe. Flow in a pipe becomes turbulent either because of high velocity, because of large pipe diameter, or because of low viscosity.
Methods for Developing Flow Patterns (i.e., finding the velocity field) Analytical Methods: The governing equations (mostly the Navier-Stokes equation) are non-linear. Closed-form solutions to these equations only exist for special, strongly simplified cases. Computational Methods: The authors of the book seem to feel that this is not overly useful. As a numerical modeler I disagree. For experimental methods, it is often difficult to find materials that scale properly to large scales (e.g., entire oceans or the Earthʼs mantle). Experimental Methods: Very useful for complicated flows, especially flows that involve turbulence. While the correct physical laws are known, time and space resolution make turbulence tricky in numerical models. Pathline, Streakline, and Streamline Important concepts in flow visualization: In steady flows, all 3 are the same. Pathline: The line (or path) that a given fluid particle takes. Streakline: The line formed by all fluid particles that have passed through a given fixed point. Streamline: A continuous line that is tangent to the velocity vectors everywhere along its path (at a given moment in time). Acceleration: Normal and Tangential Components Velocity can be written as: where V(s,t) is the speed and is a unit vector tangential to the velocity. xxxxx xxxxx The derivative of the speed is (since ds/dt = V): Streakline at t = t 0 Streakline at t > t 0
The time derivative of the unit vector is non-zero because the direction of the unit vector changes. The centripetal acceleration is Acceleration in Cartesian Coordinates This is probably one of the most fundamental concepts of the course, but it is not very intuitive! so that the total acceleration becomes or From last page, we had For a simpler example, letʼs look at the material derivative of temperature T in one dimension: T(x,t) This derivative is called the full derivative or material derivative. It is often written D/Dt instead of d/dt. It can apply to other quantities as well. The full time derivative describes the change in time of a certain property as we move along with the fluid. At a fixed point x 0, a change in temperature can be caused by two different mechanisms: 1) The temperature of the local fluid particle changes (e.g., due to heat conduction, radioactive heating, etc ): 2) All fluid particles keep their temperature, but the velocity u brings a new particle to x 0 which has a different temperature:
The changes are called local change and convective change (the convective change is also called advective change ) Example 4.1 (p. 94): Find the acceleration half-way through the nozzle Velocity is given as = local temperature change = local acceleration in x = convective temperature change = convective acceleration in x Taking the x-derivative of u, and multiplying it times u gives: Just plug in the values and x = 0.5L to get the answer Lagrangian reference frame: In the Lagrangian frame (moving along with a fluid particle), there is no convective acceleration. Because youʼre staying with a given particle, no other particle can come in and bring with it a different velocity. Eulerian: (same in x and y) The convective terms may be seen as a correction due to the fact that new particles with different properties are moving into our observation volume. Note that conservation laws naturally apply in the Lagrangian frame: A conserved quantity such as total energy E remains constant in a given material volume. An Eulerian observer sees different material volumes flow past, each of them possibly with different E.
Eulerʼs Equation Eulerʼs equation is Newtonʼs 2nd Law applied to a continuous fluid. Recall Newtonʼs 2nd law for a particle (balance in l-direction): This law is fundamental and thus also applies to fluid particles: (There may be additional forces ) Now we shrink the fluid element to Uniform acceleration of a tank of liquid (Fig. 4.13) (partial derivative since p may be a function of other coord.s and time) and so that Horizontal balance: Eulerʼs equation (force balance in a moving fluid) Vertical balance: (hydrostatic!)
Derivation of the Bernoulli Equation: Start with Eulerʼs equation applied along a pathline: Recall what we just did to get the Bernoulli equation: 1) Assume steady flow (donʼt apply this to anything else!) 2) Integrate forces (per volume) along a pathline. Assume steady flow ( ); all s-derivatives are now full derivatives. Integrating with respect to s, we get Bernoulliʼs eqn. The integral of force along a distance gives us energy: Note that the velocity term is the kinetic energy per unit volume. Also note that energy / volume has same units as pressure. The kinetic energy / volume is also known as kinetic pressure. Along a streamline in steady, inviscid flow, the sum of piezometric pressure plus kinetic pressure is constant. Application of the Bernoulli Equation: Stagnation Tube From the geometry of the flow we know (we observe or else we just assume; we havenʼt really derived this) that at both points there is no vertical acceleration. The vertical balance is thus hydrostatic: Apply to points 1 and 2 (same depth z): Point 2 is a stagnation point (velocity is zero) The stagnation tube is a simple device for measuring velocity.
Notice that we had to have a free surface at the top of the fluid. If the flow is in a pressurized pipe, we donʼt know whether the piezometric head measured by l is due to static pressure in the pipe or due to flow. Hence the Pitot tube Pitot tube: Look carefully; the Pitot tube is really 2 tubes in 1. If we know the static pressure at point 1 and the dynamic pressure at point 2, we have all we need to find the velocity. For more details see the book, p. 102. Rotation and Vorticity Rotation of fluid element in flow between moving and stationary parallel plates You can think of the cruciforms as small paddle wheels that are free to rotate about their center. Rotation of a fluid element in a rotating tank of fluid (solid body rotation). If the paddle wheel rotates, the flow is rotational at that point.
The net rate of rotation of the bisector is As And similarly The rotation rate we just found was that about the z-axis; hence, we may call it The property more frequently used is the vorticity and similarly The rate-of-rotation vector is Irrotational flow requires (i.e., for all 3 components)
Vortices Vortex with irrotational flow (free vortex): A vortex is the motion of many fluid particles around a common center. The streamlines are concentric circles. Choose coordinates such that z is perpendicular to the flow. In polar coordinates, the vorticity is (see p. 104 for details) (V is function of r, only) Solid body rotation (forced vortex): or A paddle wheel does not rotate in a free vortex! Forced vortex (interior) and free vortex (outside): Good approximation to naturally occurring vortices such as tornadoes. We can find the pressure variation in different vortices In general: 1) Solid body rotation: Eulerʼs equation for any vortex: 2) Free vortex (irrotational):
Application to forced vortex (solid body rotation): with Pressure as function of z and r p = 0 gives free surface