Flowing Fluids ( 유체의 흐름 ) Fluid kinematics ( 유체운동학 ) – describes the motion of fluid elements such as translation ( 이동 ), transformation ( 변형 ), rotation ( 회전 ), etc. via geometrical analysis Two ways to describe fluid motion (continuum) Lagrangian approach – describes the fluid element’s motion with time by traveling along with the fluid element, function of time Eulerian approach – describes the fluid’s motion at a fixed location, function of time and space

Flow lines ( 유동선 ) Useful for describing or visualizing fluids in motion Streamline ( 유선 ): Local velocity vector is tangent to this line at every point along the line and a given time. The tangent of the streamline → the direction of the velocity at that specific point. Very effective to describe the geometry of flow (useful for Eulerian approach) It does NOT indicate the magnitude of the velocity. An imaginary line ( 가상의 선 ) Pathline ( 유적선 ): Defines the path that a given particle of fluid has taken in a given time. Useful for Lagrangian approach. ( 입자위치의 궤적 ) Streakline ( 유맥선 ): Lines represents the points that all particles have passed prior to a given instance. ( 유체의 지나간 자취 )

Streamline, pathline, and Streakline

Flow patterns ( 유체유동의 분류 ) Uniform flow ( 등류 ) – the velocity does not change along a streamline Non-uniform flow ( 부등류 )- the velocity changes along a streamline Steady flow ( 정상류 ) – the velocity at a given point on a streamline does not change with time Unsteady flow ( 비정상류 ) – the velocity change with time Laminar flow ( 층류 ) – fluid elements in flow make layers and no exchange between the layers (Re <2000) Turbulent flow ( 난류 ) – very irregular flow given by intense mixing and irregular motion of fluid elements (Re > 2000) 1, 2, 3 dimensional flows

Uniform flow patterns. (a) Open-channel flow. (b) Flow in a pipe. Flow patterns for nonuniform flow. (a) Converging flow. (b) Vortex flow.

Laminar and turbulent flow in a straight pipe. (a) Laminar flow. (b) Turbulent flow.

Acceleration ( 가속도 ) The rate of change of the particle’s velocity with time Acceleration consists of normal component (a n, centripetal acceleration) + tangential component (a t )

Acceleration ( 가속도 ) Cartesian coordinates Remark Substantive/material derivative ( 실질 / 물질 도함 수 ) Convective accel. ( 대류가속도 )Local accel. ( 국부가속도 )

Class quiz) The following velocity field is given for a fluid in motion. Acceleration in the x, y, z directions at a point (1,2,2) and time =1 (in m/s 2 )? We have a nozzle in which the velocity of a passing fluid can be defined as where u 0 is the entrance velocity (10m/s), L is the nozzle length (0.5m) the exit velocity is 20 m/s Acceleration at the middle of the nozzle? (assume that the velocity is uniform across each section)

Euler’s equation ( 오일러 방정식 ) For a flowing fluid at a uniform acceleration Assumption: ideal fluids – inviscid (no viscous effects), incompressible, and steady flow Free body of a fluid element being accelerated in the l direction Note) If no acceleration, the pressure distribution in that direction follows hydrostatic (p+  z=const.). In the parallel and straight streamlines, the pressure in the direction normal to the streamline is also hydrostatic.

What does the Euler’s eqn. tell us? A flowing fluid possessing acceleration has a pressure distribution as a function of the acceleration applied on the fluid. The piezometric pressure is inversely proportional to the acceleration along the direction of the acceleration (At the same elevation, the pressure decreases/increases as the acceleration increases/decreases) If no acceleration, the pressure distribution in that direction follows hydrostatic (p+  z=const.). In the parallel and straight streamlines, the pressure in the direction normal to the streamline is also hydrostatic.

Application of the Euler’s eqn. Class quiz) A tank is filled with gasoline (  = 6.6 kN/m 3 ). If this tank decelerates at 3.05 m/sec 2, how much of pressure will be applied on the front bottom of the tank? Length = 6.1 m Height = 1.83 m

Bernoulli equation ( 베르누이 방정식 ) Assumption: ideal fluids – inviscid (no viscous effects), incompressible, steady, and irrotational flow, along the same streamline By integrating the Euler’s eqn. along the streamline, piezometric pressurekinetic pressurepiezometric headvelocity head The sum of pressure energy, potential energy, and kinetic energy per unit weight (pressure head, potential head, and velocity head) is always constant.

Application of the the Bernoulli eqn. For measurement of the fluid velocity Stagnation tube Pitot tube.

Rotational flow A fluid in curved motion is subject to deformation and rotation. The bisect ( 이등분선 ) of the lines consisting two mutually perpendicular faces of a fluid element rotates → rotational flow

Flows rotating with circular streamline Rotational flow – forced vortex Irrotational flow – free vortex

Pressure distribution in rotating flows From Euler’s eqn. in the direction normal to the streamline and outward from the center

Class quiz) We have a U tube containing water as shown below. If the tube rotates at a rate of 8 rad/sec around the axis that deviates from the center, how will the water be distributed in the tube?

Pressure distribution in irrotational flows From Euler’s eqn. in the direction normal to the streamline and outward from the center The Bernoulli eqn. is valid everywhere in the flow field if the flow is ideal and irrotational

Pressure distribution around a body Pressure coefficient (non-dimensional) at stagnant point, C p =1

Separation