MEC320 Tutorial 1 Governing eqs

What is the difference between Newtonian and non-Newtonian fluids?

• Newtonian: Constant viscosity; shear stress ∝ shear rate (e.g. water, air).

• Non-Newtonian: Variable viscosity; does not follow Newton’s law (e.g. ketchup, blood)

Compressible vs. Incompressible Flow

• Compressible: Density changes significantly (e.g. high-speed gas).

• Incompressible: Density is constant (e.g. liquid flow, low-speed air).

What are the different engineering approaches available to analyse a fluid problem and which approach is the most useful one?

• Analytical: Exact math solutions; only for simple cases.

• Experimental: Real-world testing; accurate but costly.

• CFD: Computer simulation; flexible and widely used.

Most Useful: CFD, for complex, cost-effective, and fast analysis.What is CFD?

What is CFD?

Computational Fluid Dynamics: A numerical method to simulate fluid flow using computers by solving discretized fluid equations.

First Step in CFD & Importance

• Step: Geometry creation + meshing.

• Importance: Defines the physical domain; affects accuracy and computation time.

Physical Principles in Fluid Analysis

• Mass conservation

• Momentum conservation (Newton’s 2nd Law)

• Energy conservation (First Law of Thermodynamics

What is a closed or analytical solution of a set of equations?

• Definition: Exact solution expressed in mathematical form (e.g. formula).

• Used for: Simple geometries and flows.

Mathematical vs. Numerical Model

• Mathematical: Uses equations to represent physical laws.

• Numerical: Solves those equations approximately using discretization.

Why is CFD considered a ‘compromise’ between having, or not having a solution of the Navier-Stokes equations?

Why: Navier-Stokes equations are too complex for exact solutions in most cases—CFD gives an approximate but practical solution.

Eulerian vs. Lagrangian Approach

• Eulerian: Observes fluid flow at fixed points in space.

• Lagrangian: Follows individual fluid particles through space and time.

What is the physical meaning of the total derivative when quantifying a change in a fluid variable such as temperature? In this context, what is the physical meaning of the local and convective derivatives?

• Meaning: Tracks how a quantity (e.g. temperature) changes for a moving fluid element.

• Local Derivative: Change at a fixed point (∂/∂t).

• Convective Derivative: Change due to movement through space (u·∇T).

What do the various terms in the continuity equation mean physically?

• ∂ρ/∂t: Rate of density change at a point.

• ∇·(ρu): Net mass flow out of a point.

• Equation: Ensures mass is conserved.

Why Use Conservation of Mass Equation?

Reason: Guarantees mass isn’t created or lost; essential for realistic fluid modeling.

Can you think about another example, where the monitored variable is velocity, to explain the physical meaning of the substantial (or total) derivative?

If fluid velocity increases as a particle moves, the total derivative of velocity shows acceleration—a combo of local and convective changes.

What is the physical meaning of velocity field with divergence zero in the incompressible case?

Meaning: Volume of fluid elements doesn’t change; no net inflow/outflow—mass is conserved with constant density.

Which of the following mathematically describe a 3D, steady-state, compressible flow?

a)\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0

b)\frac{\partial\rho}{\partial t}+\left(\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}\right)=0

c)\nabla\cdot(\rho\vec{V})=0

d)\nabla\cdot\vec{V}=0

a) and c)

(steady state:not time dependant, compressible:function of density, 3D)

Which of the following describe a 3D, incompressible flow (steady or transient)?

a)\frac{\partial\rho}{\partial t}+\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right)=0

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

c)\frac{\partial u_i}{\partial x_i}=0,\quad i=1,2

d)\nabla\cdot(\rho\vec{V})=0

b)

(incompressible:density is constant)

Which of the following describe a 2D, incompressible flow (steady or transient)?

a)\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0

b)\frac{\partial u_i}{\partial x_i}=0,\quad i=1,2

c)\nabla\cdot\vec{V}=\text{div}\vec{V}=0

d)\frac{\partial \rho}{\partial t} + \left( \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} \right) = 0

a), b), c)

(incompressible:no density)

Which of the following describe a 2D, transient, compressible flow?

a)\frac{\partial(\rho u_i)}{\partial x_i}=0,\quad i=1,2

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

c)\frac{\partial\rho}{\partial t}+\left(\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}\right)=0

d)\frac{\partial\rho}{\partial t}+u\frac{\partial\rho}{\partial x}+v\frac{\partial\rho}{\partial y}+\rho\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0

b), c), d)

(a) implies steady flow not transient)

Write the continuity equation in tensor (Einstein) notation and in vector form.

Given (standard form):

\displaystyle \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0

Tensor (Einstein) notation:

{\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_i)}{\partial x_i}=0} where x1=x x2=y x3=zu1=u u2=v u3=w

Vector form:

\displaystyle \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0

Show that the continuity equation in conservation form is equivalent to the non-conservation form using tensor notation.

Conservation form (tensor):

\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_i)}{\partial x_i} = 0

use product rule:

\displaystyle \frac{\partial (\rho u_i)}{\partial x_i} = u_i \frac{\partial \rho}{\partial x_i} + \rho \frac{\partial u_i}{\partial x_i}

Sub into original equation:

\displaystyle \frac{\partial \rho}{\partial t} + u_i \frac{\partial \rho}{\partial x_i} + \rho \frac{\partial u_i}{\partial x_i} = 0

Group terms:

\frac{D \rho}{D t} + \rho \nabla \cdot \vec{u} = 0

Which gives the non-conservation form (substantial derivative):

\frac{D \rho}{D t} + \rho \nabla \cdot \vec{u} = 0

What types of forces can act on a fluid element?

• Body forces (e.g. gravity, electromagnetic)

• Surface forces:
   • Normal stresses (pressure + viscous)
   • Shear stresses (viscous only)

What are the Stokes relationships?

They relate normal viscous stresses to shear viscosity in Newtonian fluids, assuming zero bulk viscosity:

\tau_{xx} = 2\mu \frac{\partial u}{\partial x} - \frac{2}{3} \mu \nabla \cdot \vec{u}

similar for \/

\tau_{yy}, \tau_{zz}

What variables influence shear or normal stresses?

• Velocity gradients

• Fluid viscosity

• For compressible flow: bulk viscosity and divergence of velocity

How many momentum equations should you use in your fluid problem?

One per spatial dimension (e.g. 2D → 2 equations, 3D → 3 equations).

Why is the substantial derivative needed in momentum equations?

Because it accounts for changes due to both time and fluid motion:

\frac{D\vec{u}}{Dt} = \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla) \vec{u}

In fully developed flow in a straight cylinder along x, are

\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}

significant.

\frac{\partial u}{\partial x} = 0

flow is fully developed

\frac{\partial u}{\partial y} \neq 0

velocity varies radially

When should I consider body forces?

When gravity or external fields (e.g. EM) significantly affect flow behavior.

When should I consider normal stresses?

In compressible or high-speed flows where pressure/viscous stresses are non-negligible.

When should I consider shear stresses?

In viscous flows, near walls, or at low Reynolds numbers.

When do we need to use the energy equation?

When temperature affects flow (compressible, thermal convection, or high-speed flows).

Why do we specify boundary conditions in CFD?

To close the system of PDEs and ensure a unique, physically meaningful solution.

Can we solve the Navier-Stokes equations analytically? Justify.

Rarely. Only for simple, idealized cases. They are nonlinear and coupled.

Why are the Navier-Stokes equations difficult to solve analytically?

Due to nonlinearity from the convective term and complex geometries/boundary conditions.

What does “fully developed flow” mean?

Velocity profile no longer changes in the flow direction;

\frac{\partial \vec{u}}{\partial x} = 0

find: a) volumetric flow rate b) wall shear stress at y=0 for

u(y) = \frac{G}{2\mu} y(H - y)

a)

Q = \int_0^H u(y) , dy = \frac{G H^3}{12 \mu}

b)

\tau_w = \mu \left.\frac{du}{dy}\right|{y=0} = \mu \cdot \frac{G}{2\mu}(H - 2y) \big|{y=0} = \frac{G H}{2}

What is the difference between laminar and turbulent flow

• Laminar: smooth, orderly layers

• Turbulent: chaotic, vortical, energy cascade

What are main features of turbulent flow?

• Irregularity

• Vorticity

• Diffusivity

• 3D and time-dependent

• Energy cascade

Why examine inertial vs. viscous forces?

It defines the flow regime via Reynolds number:

\text{Re} = \frac{\rho U L}{\mu}

What other factors influence turbulence?

• Geometry

• Surface roughness

• External forcing

• Initial/boundary conditions

Differences between DNS, LES, and RANS?

• DNS (Direct numerical simulation): resolves all scales (costly)

• LES (large eddy sim): resolves large scales, models small

• RANS (Reynolds averaged Navier Stokes): averages all scales, uses turbulence models

Limitations of RANS and LES models?

• Depend on empirical models

• May fail in complex flows or near-wall regions

• LES still computationally expensive

What is the law of the wall? Why “universal”?

Describes velocity near walls in turbulent flows:

u^{+}=\frac{1}{\kappa}\ln y^{+}+Bu^{+}=\frac{1}{\kappa}\ln y^{+}+B

"Universal" because it fits many flows if non-dimensionalized

How are u+ and y+ defined and interpreted?

u^+ = \frac{u}{u_\tau}

y^+ = \frac{y u_\tau}{\nu}

They represent velocity and distance scaled by wall shear.

How fine should your mesh be to resolve the viscous sublayer?

First cell center at

y^+ \leq 1

otherwise use wall functions