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6. Basic definitions, basic equations I (4.2-4.4)

• Stationary and non-stationary flow, streamline, streamtube

• One-, two-, and three-dimensional flow• Laminar and turbulent flow• Reynolds´ number• System and control volume• Continuity equationExercises: C1, C2, C4, and C7

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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

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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 independentUniform = constant section

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Unsteady uniform flow: Q = Q(t), A = constant

Unsteady non-uniform flow: Q = Q(t), A = A(x).

Steady = time independentUniform = constant section

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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.

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

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Potential flow: Flow that can be represented by streamlines.

Streakline = path made by injected colour in a flow field

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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 45o angle and during the next second the fluid is moving downwards to the right at a 45o angle etcaccording 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?

Fig. a) Streamlines Fig. b) Particle track

A •

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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.

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LAMINAR AND TURBULENT FLOW

Laminar flow • Flow along parallel paths• Shear stress proportional to

velocity gradient (τ = μ⋅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

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Reynolds experiment

• 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

Q and P variables

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Reynolds´ number

• Reynolds generalized his results by introduction of a dimensionless number (Reynolds number):

μρ

νVDVDR == ν= μ/ρ, V=Q/A

ν = kinematic viscosityμ = dynamic viscosityD = diameter (for pipes)

e

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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 flowLower critical velocity – transition of turbulent flow to

laminar flow

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The critical Reynolds number, Rc, defining the divisionbetween laminar and turbulent flow, is very dependent onthe geometry for the flow.

• Parallel walls: Rc ≅ 1000 (using mean velocity V and spacing D)

• Wide open channel: Rc ≅ 500 (using mean velocity V and depth D)

• Flow about sphere: Rc ≅ 1 (using approach velocity V and sphere diameter D)

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C1 When 0.0019 m3/s of water flow in a 76 mm pipeline at 20°C, is the flow laminar or turbulent?

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C2 What is the maximum speed at which a spherical sand grain of diameter 0.254 mm may move through water (20°C) and the flow regime be laminar?

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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.

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CONTINUITY EQUATION

• Steady flowρ1⋅V1⋅A1 = ρ2⋅V2⋅A2 (m1 = m2)

• Incompressible flowV1⋅A1 = V2⋅A2 or Q1 = Q2 (Q = V⋅A)

V: Average velocity at a section (m/s)A: Cross-section area (m2) Q: Flow rate (m3/s)

m1 m2

Control volume

Fluid system volume

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Continuity equation applied to changingpipe diameter

Q = constant, A =(x)V1⋅A1 = V2⋅A2 or Q1 = Q2(Q = V⋅A)

Q1=V1 ⋅ A1

Q1Q2

Q2=V2 ⋅ A2

Control volume

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• Flow in a pipe junction

Q1 + Q2 = Q3 or V1⋅A1 + V2⋅A2 = V3⋅A3

• Channel flow (unsteady)

d(Vol)/dt = Q1– Q2

(Q12 = 0)

Vol: Volume of water in channel between section 1 and 2

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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?

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C7 Using the Y and the control volume in the fig. find the mixture flowrate and density if freshwater (ρ1 = 1000 kg/m3) enters section 1 at 50 l/s, while saltwater (ρ2 = 1030 kg/m3) enters section 2 at 25 l/s.