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2. MAPPING THE NUMERICAL SIMULATIONS OF PARTIAL
DIFFERENTIAL EQUATIONS
On the other hand, the development time of an optimized software solution is
much shorter than designing a reconfigurable architecture [56][57], however, the
computational efficiency is smaller in terms of area and power. Namely on FPGAs
we can make a more specific structure for the CFD with better performance in
terms of the area and dissipation with a variable accuracy considering to use it
in real life applications.
2.3.2 Fluid Flows
A wide range of industrial processes and scientific phenomena involve gas or fluids
flows over complex obstacles. In engineering applications the temporal evolution
of non-ideal, compressible fluids is quite often modeled by Navier-Stokes [58, 59]
equations. It is based on the fundamental laws of mass-, momentum- and energy
conservation, extended by the dissipative effects of viscosity, diffusion and heat
conduction. By neglecting all the above non-ideal processes, and assuming adiabatic
variations, we obtain the Euler equations [60, 61], describing the dynamics
of dissipation-free, inviscid, compressible fluids. The equations, a coupled set of
nonlinear hyperbolic partial differential equations, in conservative form expressed
as
∂ (ρv)
∂t
∂E
∂t
∂ρ
∂t
+ ∇ · (ρv) = 0 (2.2)
)
+ ∇ ·
(ρvv + Îp = 0 (2.3)
+ ∇ · ((E + p)v) = 0, (2.4)
where t denotes time, ∇ is the Nabla operator, ρ is the density, u, v are the x-
and y-components of velocity vector v, respectively, p is the pressure of the fluid,
Î is the identity matrix, and E is the total energy density defined as
E =
p
γ − 1 + 1 ρv · v, (2.5)
2
where γ is the ratio of specific heats. It is convenient to merge (2.2), (2.3) and
(2.4) into hyperbolic conservation law form in terms of U = [ρ, ρu, ρv, E] and the