3.3 Fluid Flow Equations

3.3 Fluid Flow Equations

1. Continuity Equation

• The continuity equation expresses the principle of conservation of mass for a fluid flow.

• For a fixed control volume, the general integral form is:

$\frac{\partial}{\partial t} \int_{CV} \rho \, dV + \int_{CS} \rho (\vec{V} \cdot \hat{n}) \, dA = 0$

• The first term is the rate of change of mass inside the control volume (CV).

• The second term is the net mass flow rate out through the control surface (CS).

1.1 Steady Flow Simplification

• For steady flow, the mass inside the CV is constant, so the time derivative term is zero:

$\int_{CS} \rho (\vec{V} \cdot \hat{n}) \, dA = 0$

• This means the net mass flow rate into the CV is zero (mass in = mass out).

1.2 Steady, One-Dimensional Flow in a Duct

• For a duct with a single inlet (1) and single outlet (2):

$\dot{m} = \rho_1 A_1 V_1 = \rho_2 A_2 V_2 = \text{constant}$

• Here, $\dot{m}$ is the mass flow rate (kg/s).

• $A$ is the cross-sectional area.

• $V$ is the average velocity normal to the area.

1.3 Incompressible Flow Simplification

• For an incompressible fluid, density $\rho$ is constant.

• The continuity equation simplifies to conservation of volumetric flow rate $Q$:

$Q = A_1 V_1 = A_2 V_2 = \text{constant}$

1.4 Differential Form (At a Point)

• For an infinitesimal fluid element, the continuity equation is:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$

• For incompressible flow ($\rho = \text{constant}$), this reduces to:

$\nabla \cdot \vec{V} = 0$

• In Cartesian coordinates, the incompressible condition is:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$

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2. Euler Equation

• The Euler equation is Newton's second law applied to an inviscid (frictionless) fluid element.

• It describes the balance between inertial, pressure, and body forces for an ideal fluid.

2.1 Form Along a Streamline

• For flow along a streamline (s-direction), the Euler equation is:

$\rho \frac{DV}{Dt} = -\frac{\partial p}{\partial s} - \rho g \frac{\partial z}{\partial s}$

• The term $\frac{DV}{Dt}$ is the total (material) acceleration:

$\frac{DV}{Dt} = \frac{\partial V}{\partial t} + V \frac{\partial V}{\partial s}$

• The term $-\frac{\partial p}{\partial s}$ is the net pressure force per unit volume.

• The term $-\rho g \frac{\partial z}{\partial s}$ is the component of the gravitational body force along the streamline.

2.2 Vector Form

• The general vector form of the inviscid momentum equation is:

$\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g}$

• This is a simplified version of the Navier-Stokes equations with viscosity set to zero.

2.3 Significance

• The Euler equation is the fundamental starting point for deriving Bernoulli's equation.

• It applies only to ideal (inviscid) fluids and neglects all frictional effects.

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3. Bernoulli’s Equation and Applications

3.1 Derivation and Standard Form

• Bernoulli's equation is obtained by integrating the Euler equation along a streamline for steady, incompressible, inviscid flow.

• The result is that the total mechanical energy per unit weight is constant along a streamline:

$\frac{p}{\rho g} + \frac{V^2}{2g} + z = \text{constant}$

• This is the head form. Each term has units of length (meters):

  • $\frac{p}{\rho g}$ is the pressure head.

  • $\frac{V^2}{2g}$ is the velocity head.

  • $z$ is the elevation head (potential head).

• The energy form (units of pressure) is:

$p + \frac{1}{2} \rho V^2 + \rho g z = \text{constant}$

3.2 Assumptions (CRITICAL)

Bernoulli's equation is valid only when all of the following assumptions hold:

• Steady flow.

• Incompressible flow (constant density $\rho$).

• Inviscid flow (no friction, no shear stresses).

• Flow along a single streamline (the constant may differ between streamlines unless the flow is also irrotational).

• No shaft work (no pumps or turbines between the two points).

• No heat transfer.

3.3 Extended Bernoulli Equation (For Real Flows)

• For real, viscous flows with pumps, turbines, and friction, the equation is extended:

$\frac{p_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_t + h_L$

• $h_p$ is the head added by a pump (mechanical energy input).

• $h_t$ is the head extracted by a turbine (mechanical energy output).

• $h_L$ is the head loss due to friction and minor losses between points 1 and 2.

3.4 Applications

Venturi Meter

• A device used to measure flow rate in a pipe.

• Based on the principle that velocity increases and pressure decreases in a constriction.

• The flow rate $Q$ is given by:

$Q = A_2 \sqrt{ \frac{2 (p_1 - p_2) / \rho}{1 - (A_2/A_1)^2} }$

$Q = A_2 \sqrt{ \frac{2g \Delta h}{1 - (A_2/A_1)^2} }$

• Where $\Delta h$ is the manometer reading.

Orifice Flow (Torricelli's Theorem)

• For flow from a large tank through a small orifice, the exit velocity is:

$V_{jet} = \sqrt{2gH}$

• $H$ is the vertical distance from the free surface to the orifice centerline.

• The actual discharge is less due to friction and contraction:

$Q_{actual} = C_d A_{orifice} \sqrt{2gH}$

• $C_d$ is the discharge coefficient (typically 0.60-0.65 for sharp-edged orifices).

Pitot Tube

• A device used to measure flow velocity by converting kinetic energy into pressure (stagnation pressure).

• The flow velocity is found from:

$V = \sqrt{\frac{2(p_{stag} - p_{static})}{\rho}}$

• $p_{stag}$ is the stagnation (total) pressure.

• $p_{static}$ is the static pressure.

Siphon

• Bernoulli's equation is used to calculate the flow rate through a siphon and to find the minimum pressure point (to check for cavitation).

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4. Momentum Equation

4.1 Principle

• The linear momentum equation for a fluid is derived from Newton's second law.

• For a fixed control volume, it states: The sum of external forces acting on the CV equals the rate of change of momentum inside the CV plus the net momentum flux out of the CV.

4.2 General Integral Form

• The Reynolds Transport Theorem gives the following form:

$\sum \vec{F} = \frac{\partial}{\partial t} \int_{CV} \vec{V} \rho \, dV + \int_{CS} \vec{V} \rho (\vec{V} \cdot \hat{n}) \, dA$

• $\sum \vec{F}$ is the vector sum of all external forces (body and surface) acting on the fluid in the CV.

• The first term on the right is the rate of change of momentum within the CV (unsteady term).

• The second term is the net momentum flux out of the control surface.

4.3 Steady Flow Simplification

• For steady flow, the momentum within the CV is constant, so the time derivative is zero:

$\sum \vec{F} = \int_{CS} \vec{V} \rho (\vec{V} \cdot \hat{n}) \, dA$

• "Forces on CV = Momentum flux out - Momentum flux in."

4.4 Steady, One-Dimensional Application

• For a duct with well-defined inlets and outlets, the equation is applied component-wise (e.g., in the x-direction):

$\sum F_x = \dot{m} (\beta_2 V_{2x} - \beta_1 V_{1x})$

• $\dot{m}$ is the mass flow rate.

• $V_{1x}$ and $V_{2x}$ are the average velocity components in the x-direction at inlet and outlet.

• $\beta$ is the momentum correction factor (often $\beta \approx 1$ for turbulent flow).

4.5 Types of Forces

• Surface Forces:

  • Pressure forces acting on the control surfaces.

  • Forces exerted by solid boundaries (walls, vanes, blades) on the fluid. The reaction to this is often the unknown we solve for.

• Body Forces: The most common is gravity (weight of fluid in the CV).

4.6 Application Procedure

1. Select a suitable fixed control volume that cuts through the fluid and exposes the unknown forces.

2. Draw all external forces acting on the fluid inside the CV.

3. Apply the momentum equation in the relevant coordinate directions.

4. Use the continuity equation and Bernoulli equation (if applicable) to find unknown velocities and pressures.

5. Solve for the unknown reaction forces (e.g., force on a bend, force on a gate).

4.7 Common Applications

• Calculating the anchoring force required to hold a pipe bend or nozzle in place.

• Determining the force exerted by a fluid jet on a stationary or moving flat or curved vane (fundamental for turbine analysis).

• Analyzing the force on a sluice gate.

• Calculating the drag force on a body immersed in a flow.

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5. Dimensional Analysis and Similitude

5.1 Purpose of Dimensional Analysis

• To reduce the number of variables in a physical problem by grouping them into dimensionless parameters.

• To guide experimental work and establish similitude (scaling laws) between models and prototypes.

• To present data in a compact, universal form.

5.2 Fundamental Dimensions

• In fluid mechanics, the primary dimensions are Mass (M), Length (L), and Time (T).

• Examples:

  • Velocity: $L T^{-1}$

  • Acceleration: $L T^{-2}$

  • Force: $M L T^{-2}$

  • Pressure: $M L^{-1} T^{-2}$

  • Viscosity: $M L^{-1} T^{-1}$

5.3 Buckingham Pi Theorem

• Statement: If a physical process involves $n$ variables and $k$ fundamental dimensions, it can be described by $(n - k)$ independent dimensionless Pi terms.

• Procedure:

  1. List all $n$ physical variables that affect the phenomenon.

  2. Express each variable in terms of the fundamental dimensions (M, L, T).

  3. Determine the number of fundamental dimensions involved ($k$).

  4. Choose $k$ repeating variables that collectively contain all fundamental dimensions (e.g., length $L$, velocity $V$, density $\rho$).

  5. Form each Pi term as a product of the non-repeating variables and the repeating variables raised to unknown exponents.

  6. Solve for the exponents by making each Pi term dimensionless.

5.4 Common Dimensionless Numbers

Reynolds Number (Re)

• Ratio of inertial forces to viscous forces.

$Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}$

• Determines the flow regime (laminar or turbulent).

• Dominant in pipe flow, flow around submerged objects.

Froude Number (Fr)

• Ratio of inertial forces to gravitational forces.

$Fr = \frac{V}{\sqrt{g L}}$

• Dominant in free-surface flows where gravity is important (open channels, ship waves, spillways).

Mach Number (Ma)

• Ratio of flow speed to the speed of sound.

$Ma = \frac{V}{c}$

• Dominant in compressible, high-speed flows (aircraft, gas turbines).

Euler Number (Eu)

• Ratio of pressure forces to inertial forces.

$Eu = \frac{\Delta p}{\rho V^2}$

• Relates pressure drop to flow velocity.

Weber Number (We)

• Ratio of inertial forces to surface tension forces.

$We = \frac{\rho V^2 L}{\sigma}$

• Important in flows with interfaces, droplets, and bubbles.

5.5 Similitude (Model Scaling)

• Similitude means similarity between a model and its prototype.

• Dynamic similarity exists when all relevant dimensionless numbers are the same for the model and the prototype.

Types of Similarity

• Geometric Similarity: Model and prototype have the same shape (all linear dimensions have a constant scale ratio $L_r$).

• Kinematic Similarity: Velocities at corresponding points are in a constant ratio (velocity ratio $V_r$ is constant).

• Dynamic Similarity: Force ratios (dimensionless numbers) are identical.

Scaling Laws

• Reynolds Similitude: Matching Reynolds number is required for flows dominated by viscous effects (e.g., pipe flow, submarine motion).

$Re_m = Re_p \quad \Rightarrow \quad \frac{V_m L_m}{\nu_m} = \frac{V_p L_p}{\nu_p}$

• Froude Similitude: Matching Froude number is required for free-surface flows dominated by gravity (e.g., ship hulls, spillways).

$Fr_m = Fr_p \quad \Rightarrow \quad \frac{V_m}{\sqrt{g L_m}} = \frac{V_p}{\sqrt{g L_p}} \quad \Rightarrow \quad V_r = \sqrt{L_r}$

• Mach Similitude: Matching Mach number is required for high-speed compressible flows.

5.6 Scale Effects

• It is often impossible to satisfy all similitude conditions simultaneously (e.g., matching both Re and Fr for a ship model).

• The engineer must match the most dominant dimensionless number and then correct for the effects of mismatched parameters.

5.7 Applications

• Designing scale models for wind tunnel testing (aircraft, buildings).

• Towing tank testing of ship hulls.

• Hydraulic model studies of dams, spillways, and harbors.

• Correlating experimental data for pipe friction, drag coefficients, and pump/turbine performance curves.