Faros University ME 253 Fluid Mechanics II Fluid Kinematics Dr. A. Shibl Text Book: Frank White “ Fluid Mechanics” Flow Classification • • • • • • Real (Viscous) or Ideal (Inviscid) Flow Compressible or Incompressible Flow Turbulent or Laminar Flow Unsteady or Steady Flow 3, 2, or 1 dimensional Flow Rotational or Iroratational Flow Fluid Kinematics: Introduction • • • • Fluids subject to shear, flow Fluids subject to pressure imbalance, flow Kinematics : Description of Fluid motion. Visualization. Fluid Motion • Two ways to describe fluid motion – Lagrangian • Follow particles around – Eularian V dx dy dz i j k dt dt dt • Watch fluid pass by a V ui vj wk point or an entire region – Flow pattern • Streamlines – velocity is tangent to them 4 STEADY AND UNSTEADY FLOW: Steady flow: the flow in which conditions at any point do not change with time is called steady P V flow. 0, 0, 0, t t t • Unsteady flow: the flow in which conditions at any point change with time, is called unsteady flow. P V t 0, t 0, t 0, 5 UNIFORM AND NON-UNIFORM: • The flow in which the conditions at all points are the same at the same instant is uniform flow. P V 0, 0, 0, s s s • The flow in which the conditions navy from point to point at the same instant is non-uniform flow. P V 0, 0, 0, s s s 6 Velocity Field Velocity, acceleration, and pressure of these particles at a given time are: Velocity Field: Eulerian and Lagrangian Eulerian: the fluid motion is given by completely describing properties as a function of space and time. We obtain information about the flow by noting what happens at fixed points. Lagrangian: following individual fluid particles as they move about and determining how the fluid properties of these particles change as a function of time. Measurement of Temperature Eulerian Lagrangian Eulerian methods are commonly used in fluid experiments or analysis—a probe placed in a flow. Lagrangian methods can also be used if we “tag” fluid particles in a flow. Velocity Field: 1D, 2D, and 3D Flows Three-Dimensional Flow: All three velocity components are important and of equal magnitude. Flow past a wing is 3D flow Two-Dimensional Flow: If one of the velocity components is small relative to the other two, thus it is reasonable to assume 2D flow. One-Dimensional Flow: If two of the velocity components may be small relative to the other one, thus it is reasonable to assume 1D flow. Steady Flow: The velocity at a given point in space does not vary with time. Velocity Field: Streamlines Streamline: the line that is everywhere tangent to the velocity field. Analytically, for 2D flows, integrate the equations defining lines tangent to the velocity field: ACCELERATION • Acceleration = rate of change of velocity • Components: – Normal – changing direction – Tangential – changing speed V V ( s , t )et det dV dV a et V dt dt dt dV V V V dt s t det V en dt r V V V 2 a (V )et en s t r 11 Acceleration Field Lagrangian Frame: Eulerian Frame: we describe the acceleration in terms of position and time without following an individual particle. A fluid particle can accelerate due to a change in velocity in time (“unsteady”) or in space (moving to a place with a greater velocity). Acceleration Field time dependence spatial dependence We note: Then, substituting: The above is good for any fluid particle, so we drop “A”: Acceleration Field Writing out these terms in vector components: x-direction: y-direction: z-direction: Writing these results in “short-hand”: where, () ˆ ˆ ˆ i j k , x y z Acceleration Field: Unsteady Effects Consider flow in a constant diameter pipe, where the flow is assumed to be spatially uniform: 0 0 0 0 0 Acceleration Field: Convective Effects The portion of the material derivative represented by the spatial derivatives is termed the convective term or convective accleration: It represents the fact the flow property associated with a fluid particle may vary due to the motion of the particle from one point in space to another. Convective effects may exist whether the flow is steady or unsteady. Example 1: Example 2: Acceleration = Deceleration ACCELERATION • Cartesian coordinates V ui vj wk a ax i a y j az k du u dx u dy u dz u u u u u . u v w dt x dt y dt z dt t x y z t dv v dx v dy v dz v v v v v ay . u v w dt x dt y dt z dt t x y z t dw w dx w dy w dz w w w w w az . u v w dt x dt y dt z dt t x y z t ax • • • • • In steady flow ∂u/∂t = 0 , local acceleration is zero. Convective In unsteady flow ∂u/∂t ≠ 0 ; local acceleration Occurs. Other terms u ∂u/∂x, v ∂u/∂y,.. are called convective accelerations. Convective acceleration Occurs when the velocity varies with position. Uniform flow: convective acceleration = 0 Non-uniform flow: convective acceleration ≠0 Local 17 Example 2 Given : V 3ti xzj ty k Find : Acceleration, a u 3t ; v xz; w ty 2 u u u u u v w 0(3t ) 0( xz ) 0(ty 2 ) 3 3 x y z t v v v v u v w z (3t ) 0( xz ) x(ty 2 ) 0 3 zt xy 2t x y z t w w w w u v w 0(3t ) 2ty( xz ) 0(ty 2 ) y 2 2 xyzt y 2 x y z t ax ay az 2 2 a a x i a y j a z k 3i (3tz txy ) j (2 xyzt y )k 18