# 6. Basic basic equations I ( )

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1 6. Basic basic equations I ( ) Steady and uniform flows, streamline, streamtube One-, two-, and three-dimensional flow Laminar and turbulent flow Reynolds number System and control volume Continuity equation Exercises: C1, C2, C4, and C7

2 CLASSIFICATION OF FLOWS Flow characterized by two parameters time and distance. Division of flows with respect to time: Steady flow time independent Unsteady flow time dependent Quasi-steady flow slow changes with time Division of flows with respect to distance: Uniform flow constant section area along flow path Non-uniform flow variable section area

3 Examples of flow types: Steady uniform flow: flowrate (Q) and section area (A) are constant Steady non-uniform flow: Q = constant, A = A(x). Steady = time independent Uniform = constant section

4 Unsteady uniform flow: Q = Q(t), A = constant Unsteady non-uniform flow: Q = Q(t), A = A(x). Steady = time independent Uniform = constant section

5 VISUALIZATION OF FLOW PATTERNS Streamline: a curve that is drawn in such a way that it is tangential to the velocity vector at any point along the curve. A curve that is representing the direction of flow at a given time. No flow across a stream line.

6 Streamtube: A set of streamlines arranged to form an imaginary tube. Ex: The internal surface of a pipeline

7 Potential flow: Flow that can be represented by streamlines. Streakline = path made by injected colour in a flow field

8 Example streamline and streakline A flowfield is periodic in such a way that the streamline pattern is repeated at fixed intervals. During the first second the fluid is moving upwards to the right at a 45 o angle and during the next second the fluid is moving downwards to the right at a 45 o angle etc according to Fig. a). The flow velocity is constant = 10 m/s. After 2.5 s the particle track for a particle that is released at point A at time zero is shown in Fig. b). If colour is injected continuously at point A from time 0 how will the resulting streakline look like after 2.5 s? A Fig. a) Streamlines Fig. b) Particle track

9 TWO WAYS OF DESCRIBING FLUID MOTION Lagrangian view: the path, density, velocity and other characteristics of each fluid particle in a flow is traced. Eulerian view: study the flow characteristics (velocity, pressure, density, etc.) and their variation with time at fixed points in space.

10 LAMINAR AND TURBULENT FLOW Laminar flow Flow along parallel paths Shear stress proportional to velocity gradient ( = du/dy) Disturbances in the flow are rapidly damped by viscous action Turbulent flow Fluid particles moves in a random manner and not in layers Length scales >> molecular scales in laminar flow Rapid continuous mixing Inertia forces and viscous forces of importance

11 Reynolds experiment Q and P variables Small velocities line of dye intact, movement in parallel layers laminar flow High velocities rapid diffusion of dye, mixing turbulent flow Critical velocity line of dye begin to break-up, transition between laminar and turbulent flow

12 Reynolds number Reynolds generalized his results by introduction of a dimensionless number (Reynolds number): R e VD VD = /, V=Q/A = kinematic viscosity = dynamic viscosity D = diameter (for pipes)

13 Reynolds numbers for pipe flow Laminar flow Re < 2000 Transitional flow Re = 2000 to 4000 Turbulent flow Re > 4000 Two thresholds: Upper critical velocity transition of laminar flow to turbulent flow Lower critical velocity transition of turbulent flow to laminar flow

14 The critical Reynolds number, R c, defining the division between laminar and turbulent flow, is very dependent on the geometry for the flow. Parallel walls: R c 1000 (using mean velocity V and spacing D) Wide open channel: R c 500 (using mean velocity V and depth D) Flow about sphere: R c 1 (using approach velocity V and sphere diameter D)

15 C1 When m 3 /s of water flow in a 76 mm pipeline at 20 C, is the flow laminar or turbulent?

16

17 C2 What is the maximum speed at which a spherical sand grain of diameter mm may move through water (20 C) and the flow regime be laminar?

18 FLUID SYSTEM AND CONTROL VOLUME Fluid system: Specified mass of fluid within a closed surface Control volume: Fix region in space that can t be moved or change shape. Its surface is called control surface.

19 CONTINUITY EQUATION Steady flow 1 V 1 A 1 = 2 V 2 A 2 (m 1 = m 2 ) m 1 m 2 Incompressible flow V 1 A 1 = V 2 A 2 or Q 1 = Q 2 (Q = V A) Control volume Fluid system volume V: Average velocity at a section (m/s) A: Cross-section area (m 2 ) Q: Flow rate (m 3 /s)

20 Continuity equation applied to changing pipe diameter V 1 A 1 = V 2 A 2 or Q 1 = Q 2 (Q = V A) Q = constant, A =(x) Q 1 Q 2 Q 1 =V 1 A 1 Q 2 =V 2 A 2 Control volume

21 Flow in a pipe junction Channel flow (unsteady) d(vol)/dt = Q 1 Q 2 (Q 12 = 0) Q 1 + Q 2 = Q 3 or Vol: Volume of water in channel between section 1 and 2 V 1 A 1 + V 2 A 2 = V 3 A 3

22 C4 Water flows in a pipeline composed of 75 mm and 150 mm pipe. Calculate the mean velocity in the 75 mm pipe when that in the 150 mm pipe is 2.5 m/s. What is its ratio to the mean velocity in the 150 mm pipe?

23 C7 Using the Y and the control volume in the fig. find the mixture flowrate and density if freshwater ( 1 = 1000 kg/m 3 ) enters section 1 at 50 l/s, while saltwater ( 2 = 1030 kg/m 3 ) enters section 2 at 25 l/s.

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