In the previous lessons we have derived the governing equations for fluid dynamics,
the Navier-Stokes equations. The original Navier-Stokes equations
from the eighteen forty’s only included mass and momentum equations while today we also consider
the conservation of energy into the equations set. In general the Navier-Stokes equations are
five coupled nonlinear partial differential equations that are solved for five unknowns:
the tree velocity components and two of the state variables such as pressure density or energy,
that are connected by an equation of state. So now that we have a set of equations how do
we apply them to solve real world problems? A mathematical model requires five basic elements:
first is the problem statement. We need to define what is the problem that we want to
solve. Second we introduce modeling assumptions so that simplifies the problem and we modify the
model equations accordingly. In order to close the model we need to describe the flow conditions on
its boundaries introducing boundary conditions and if the problem is unsteady the initial
condition of the domain is required. These five elements will let us obtain a well posed problem
that will provide an unique solution. Let's analyze each one of these components in
detail. Let's start with the problem statement. We want to define what is the problem we want to
analyze. So first we need to define the geometry the fluid flow problem and its
boundaries. Let's consider a racing car, we first define the geometry of the car then we define the
flow field that surrounds the car. We define the properties of the fluid such as density,
viscosity and so on. Next we define the flow conditions, for example the velocity of the flow,
its pressure and temperature. Last, we define what the goals for our analysis are. In this example,
we want to analyze the downforce generated by the front and the rear wing and the drag force acting
on the car in order to improve the aerodynamics properties of the car and win the race.
Once the problem statement is defined we can introduce modeling assumptions
based on the phenomena characteristics in order to simplify the model. For example,
a water jet that exits the nozzle at large speed won’t be affected by gravity during its initial
path. We can then remove the gravity effects from our analysis. Another assumption could
be to simplify the analysis moving from a full 3D model into a 2D version of it. For example,
to get a quick estimate of the downforce generated by the spoiler of the sport car we saw earlier,
we can just take a section of the wing and analyze a 2D version if it. In this way we
can get a quick estimate of the down force generated by the entire spoiler. Based on
the assumption introduced we can then modify the Navier Stokes equation accordingly.
The last step is to define the fluid domain and the boundary and initial conditions. Let's assume
a T junction in pipe system. Our fuid domain will be the water contained in the junction.
We then need to specify a set of boundary conditions to define how the fluid will
behave. First of all, the liquid is contained by the pipes, that act as walls avoiding the fluid to
be dispersed. Now we have three sides that are still open and additional boundary conditions
must be specified. Depending on the condition specified we will have a unique solution for
the mathematical model. As you can see here, changing the direction of the fluid crossing
the boundaries we will have different solutions. - For unsteady solutions we also have to specify
an initial condition. If we assume to have cold water as initial condition and hot water entering
the system we will see the system heat up in time. Initial conditions and boundary conditions provide
a unique solution to our mathematical model. The Navier- Stokes equation are a set of nonlinear
partial differential equations or PDEs. Depending on the flow conditions and the physics that we're
analyzing the Navier-Stokes equation can be any of the PDEs conditions. Inviscid incompressible
steady state flow problems are an elliptical problem like the mixing tee we just analyzed,
under the assumption of steady state flow. In case of viscous subsonic unsteady flow,
like the exhaust jet of a commercial airplane engine we have a parabolic problem. For supersonic
flows such as the one that we have around a jet fighter moving at supersonic speed the
problem becomes hyperbolic. It is also common to have mixed problems, namely regions of fluid
in subsonic and supersonic regimes, like, for example, in the donations where a supersonic
expansion happens in quiescent air. This can be typically seen in constructions demolitions
and also in some powerful volcano eruptions. The boundary conditions for velocity at walls are
common for all problem. It has been observed and confirmed experimentally that for viscous fluids
the velocity of the fluid particles on solid boundaries is zero relative to boundary. This
is called no-slip condition. For rarefied gas flows and for inviscid flows approximations the
fluid can move tangentially to the wall. This condition is called no-penetration condition,
that states that the fluid cannot cross a solid boundary, meaning that the component of the
flow velocity normal to the solid boundary must be zero. Other boundary conditions will be introduced
for specific fluid problems as they arise. In this lecture we analyzed how to set up
our mathematical model in order to analyze a specific problem and obtain a unique solution.