3.4 Laminar and Turbulent Flow

3.4 Laminar and Turbulent Flow

Introduction to Real Viscous Flows

While inviscid flow theory (Bernoulli, Euler) provides valuable insights, all real fluids possess viscosity. This viscosity fundamentally alters flow behavior, leading to two distinct regimes: laminar and turbulent. Viscosity is responsible for generating shear stress, creating velocity gradients, dissipating energy as heat (head loss), and forming boundary layers on solid surfaces. This section explores the detailed characteristics of laminar and turbulent flows, the calculation of energy losses in piping systems, and the critical concepts of boundary layer growth and flow separation, which govern drag and pressure distribution on immersed bodies.

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1. Laminar Flow in Pipes and Plates

1.1 Characteristics of Laminar Flow

1. Motion: Fluid particles move in smooth, parallel layers or laminae with no macroscopic mixing between adjacent layers.

2. Mechanism: Momentum transfer occurs solely via molecular diffusion (viscosity). Shear stress is governed by Newton's law: $\tau = \mu (du/dy)$.

3. Stability: Laminar flow is stable to small disturbances only at low Reynolds numbers. It is the preferred state when viscous forces dominate inertial forces.

1.2 Fully Developed Laminar Flow in a Circular Pipe (Hagen-Poiseuille Flow)

1. Assumptions:

   • Steady, incompressible flow.

   • Fully developed (velocity profile doesn't change along the pipe).

   • Axisymmetric flow.

   • Constant fluid properties.

2. Velocity Profile (Parabolic): Derivation involves a force balance on a cylindrical fluid element: Pressure force = Viscous shear force. $u(r) = \frac{1}{4\mu} \left( -\frac{dp}{dx} \right) (R^2 - r^2)$

   • $u(r)$: Velocity at radial position $r$.

   • $R$: Pipe radius.

   • $dp/dx$: Constant pressure gradient (negative for flow in +x direction).

   • Maximum Velocity (at centerline, $r=0$): $u_{max} = \frac{1}{4\mu} \left( -\frac{dp}{dx} \right) R^2$

   • Average Velocity: $V_{avg} = \frac{Q}{A} = \frac{1}{2} u_{max}$

3. Volume Flow Rate (Q): Integrating the velocity profile gives the Hagen-Poiseuille Equation: $Q = \frac{\pi R^4}{8\mu} \left( -\frac{dp}{dx} \right) = \frac{\pi D^4}{128\mu} \left( -\frac{dp}{dx} \right)$ This shows flow rate is proportional to the fourth power of the radius ($R^4$), making it extremely sensitive to pipe diameter.

4. Shear Stress Distribution: $\tau(r) = \frac{r}{2} \left( -\frac{dp}{dx} \right)$ Shear stress varies linearly from zero at the centerline to a maximum at the wall. $\tau_w = \frac{R}{2} \left( -\frac{dp}{dx} \right)$

1.3 Fully Developed Laminar Flow Between Parallel Plates

1. Setup: Two infinite parallel plates separated by distance $h$, with lower plate stationary and upper plate either stationary or moving (Couette flow).

2. Velocity Profile (Parabolic for stationary plates): $u(y) = \frac{1}{2\mu} \left( -\frac{dp}{dx} \right) (hy - y^2)$

   • $y$: Distance from the lower plate.

   • Maximum Velocity (at mid-plane, $y = h/2$): $u_{max} = \frac{h^2}{8\mu} \left( -\frac{dp}{dx} \right)$

   • Average Velocity: $V_{avg} = \frac{2}{3} u_{max}$

3. Applications: Lubrication theory (thin film flows), flow in very narrow gaps.

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2. Major and Minor Losses

2.1 Energy Loss in Pipe Systems

In real viscous flows, mechanical energy is irreversibly converted to thermal energy (heat) due to friction. This appears as a head loss ($h_L$) in the Extended Bernoulli Equation. Head losses are classified as:

2.2 Major Losses (Friction Losses)

1. Definition: Head loss due to viscous friction along the straight, constant-diameter sections of pipe.

2. Darcy-Weisbach Equation: The fundamental equation for calculating major losses. $h_f = f \frac{L}{D} \frac{V^2}{2g}$

   • $h_f$: Friction head loss (m or ft of fluid).

   • $f$: Darcy friction factor (dimensionless) – the key parameter.

   • $L$: Length of pipe.

   • $D$: Pipe diameter.

   • $V$: Average flow velocity.

   • $g$: Gravitational acceleration.

3. The Friction Factor (f):

   • For Laminar Flow (Re < 2000): $f = \frac{64}{Re}$ This is derived analytically from the Hagen-Poiseuille solution.

   • For Turbulent Flow (Re > 4000): $f$ depends on Reynolds number (Re) and relative roughness ($\epsilon/D$).

     • $\epsilon$: Average height of pipe wall roughness (m).

     • The relationship $f = \phi(Re, \epsilon/D)$ is given by the Colebrook-White equation (implicit) or the Moody chart (graphical).

4. Moody Chart: A log-log plot of $f$ vs. $Re$ with $\epsilon/D$ as parameter.

   • For smooth pipes ($\epsilon/D \approx 0$), $f$ decreases with Re.

   • For rough pipes, $f$ becomes constant at high Re (fully rough regime), independent of Re.

2.3 Minor Losses (Local Losses)

1. Definition: Head loss associated with flow disturbances caused by fittings, valves, bends, expansions, contractions, entrances, and exits.

2. Cause: Flow separation, secondary flows, and increased turbulence dissipate energy.

3. Calculation: Expressed in terms of the loss coefficient (K). $h_m = K \frac{V^2}{2g}$

   • Important: The velocity $V$ in this formula is typically the velocity in the smaller-diameter pipe or in the main flow stream after the fitting, unless specified otherwise.

   • For pipe expansions/contractions, the velocity head used is that of the higher velocity stream.

4. Equivalent Length Method: Sometimes minor losses are accounted for by adding an equivalent length ($L_e$) of straight pipe that would produce the same head loss. $h_m = f \frac{L_e}{D} \frac{V^2}{2g} \quad \Rightarrow \quad L_e = \frac{K D}{f}$

5. Common Loss Coefficients (Approximate):

   • Pipe entrance (sharp-edged): $K \approx 0.5$

   • Pipe exit (into a reservoir): $K \approx 1.0$ (All kinetic energy is lost).

   • 90° standard elbow: $K \approx 0.3 - 0.9$

   • Fully open gate valve: $K \approx 0.15 - 0.2$

   • Fully open globe valve: $K \approx 10$

2.4 Total Head Loss in a System

For a series pipeline, losses are additive: $h_{L,total} = \sum h_{f,i} + \sum h_{m,j} = \sum \left( f \frac{L}{D} \frac{V^2}{2g} \right)_i + \sum \left( K \frac{V^2}{2g} \right)_j$

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3. Boundary Layer Theory

3.1 Concept and Development (Prandtl, 1904)

1. Key Insight: For high Reynolds number flows over streamlined bodies, viscous effects are confined to a thin layer very close to the solid surface—the boundary layer. Outside this layer, the flow can be treated as inviscid and irrotational.

2. Boundary Layer Characteristics:

   • No-slip condition forces velocity to be zero at the wall.

   • Velocity increases from zero at the wall to the free-stream velocity ($U$) at the edge of the boundary layer.

   • Large velocity gradients exist within the thin layer, producing significant viscous shear stress.

   • Pressure is approximately constant across the thin boundary layer (in the direction normal to the wall).

3.2 Boundary Layer Thickness Definitions

1. Boundary Layer Thickness ($\delta$): The distance from the wall where the local velocity $u = 0.99U$. (99% of free-stream).

2. Displacement Thickness ($\delta^*$):

   • Represents the amount by which the inviscid streamlines are displaced outward due to the formation of the boundary layer.

   • Due to slower flow near the wall, mass flow is reduced. $\delta^*$ is the distance the wall would need to be moved outward to have the same mass flow with a uniform velocity $U$. $\delta^* = \int_0^\infty \left( 1 - \frac{u}{U} \right) dy$

3. Momentum Thickness ($\theta$):

   • Represents the loss of momentum flux in the boundary layer compared to the inviscid flow. Related to skin friction drag. $\theta = \int_0^\infty \frac{u}{U} \left( 1 - \frac{u}{U} \right) dy$

4. Shape Factor: $H = \delta^* / \theta$. Indicates the "fullness" of the velocity profile. Laminar: $H \approx 2.6$; Turbulent: $H \approx 1.3-1.4$.

3.3 Laminar and Turbulent Boundary Layers

1. Laminar Boundary Layer:

   • Smooth, orderly growth from the leading edge.

   • Velocity profile is approximately parabolic.

   • Blasius Solution (for flat plate at zero pressure gradient): $\delta(x) \approx \frac{5.0x}{\sqrt{Re_x}}, \quad \text{where } Re_x = \frac{Ux}{\nu}$ Shear stress: $\tau_w(x) \approx \frac{0.332 \rho U^2}{\sqrt{Re_x}}$

2. Transition to Turbulence:

   • Occurs at a critical Reynolds number $Re_{x,crit} \approx 3 \times 10^5 \text{ to } 5 \times 10^6$, depending on surface roughness and free-stream turbulence.

   • Caused by instability of the laminar profile to small disturbances.

3. Turbulent Boundary Layer:

   • Characterized by chaotic, three-dimensional fluctuations and intense mixing.

   • Velocity profile is much fuller (logarithmic law of the wall).

   • Grows more rapidly than a laminar layer: $\delta(x) \approx \frac{0.37x}{Re_x^{1/5}}$.

   • Wall shear stress is significantly higher than in a laminar layer at the same Rex.

3.4 Skin Friction Drag

The net viscous force on a surface due to shear stress integrated over its area.

• For a flat plate of width b and length L: $F_D = \int_0^L \tau_w(x) b \, dx$

• Skin friction drag coefficient: $C_f = \frac{F_D}{\frac{1}{2}\rho U^2 A}$

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4. Flow Separation

4.1 Mechanism of Separation

1. Adverse Pressure Gradient: A region where pressure increases in the flow direction ($dp/dx > 0$). This occurs on the downstream side of a bluff body or in a diffuser.

2. Process:

   • Fluid particles near the wall have low momentum due to viscous friction.

   • As they move into an adverse pressure gradient, they are slowed down by both friction and the opposing pressure force.

   • Eventually, their kinetic energy is exhausted, and they come to a stop.

   • Fluid further downstream is forced to move away from the wall, creating a backward flow and causing the streamline to detach from the surface. This point is the separation point.

3. Separation Point: Defined where the wall shear stress becomes zero and reverses direction. $\left. \frac{\partial u}{\partial y} \right|_{y=0} = 0 \quad \text{at separation}$

4.2 Consequences of Separation

1. Formation of a Wake: A region of low-pressure, recirculating, and highly turbulent flow behind the body.

2. Increased Drag: Separation dramatically increases pressure drag (form drag) because the low-pressure wake creates a large net pressure force opposing motion. This often dominates over skin friction drag for bluff bodies.

3. Loss of Lift: On airfoils, separation disrupts smooth flow over the upper surface, reducing lift and causing stall.

4. Energy Loss: Significant energy dissipation occurs in the turbulent wake.

4.3 Factors Influencing Separation

1. Pressure Gradient: The stronger the adverse pressure gradient, the earlier and more severe the separation.

2. Boundary Layer Type: Turbulent boundary layers resist separation better than laminar ones because the fuller velocity profile gives near-wall fluid particles more momentum to fight the adverse gradient.

3. Surface Roughness: Can trip the boundary layer from laminar to turbulent earlier, delaying separation (this is why golf balls have dimples).

4. Body Shape (Streamlining): A streamlined body (e.g., airfoil) is designed to create a gradual pressure rise over the rear, minimizing or preventing separation. A bluff body (e.g., cylinder, flat plate normal to flow) causes abrupt separation.

4.4 Control of Separation

Engineers use various techniques to delay separation and reduce drag:

• Streamlining: Shape the body to minimize adverse gradients.

• Vortex Generators: Small fins that energize the boundary layer by bringing high-momentum fluid from the outer flow down to the wall.

• Boundary Layer Suction: Remove low-momentum fluid near the wall through porous surfaces.

• Boundary Layer Blowing: Inject high-momentum fluid into the boundary layer.

4.5 Example: Flow Over a Circular Cylinder

• At very low Re (Re < 5): No separation, creeping flow.

• Moderate Re (Re ~ 10-40): Steady separation forms two symmetric standing vortices.

• High Re (Re > 1000): Unsteady separation leads to periodic vortex shedding (Kármán vortex street), causing oscillating forces and potential structural vibration.

Conclusion: The transition from laminar to turbulent flow marks a fundamental shift in transport phenomena, dramatically affecting head loss, mixing, and drag. Boundary layer theory successfully bridges inviscid external flow with viscous wall effects, providing the framework to analyze skin friction and predict the critical phenomenon of flow separation, which is paramount in aerodynamic and hydrodynamic design. The ability to calculate major and minor losses enables the effective design of pressurized pipe networks, a cornerstone of civil engineering infrastructure.