Hydrostatic pressure arises due to the weight of a fluid acting on an object.
Hydrodynamic pressure arises due to a fluid moving against an object and any pressure gradient in the system.
These two pressures will occur together in real flows.
Fluid flow, as it is derived from the main equations of fluid dynamics, is often discussed in terms of flow rates or forces. However, we sometimes like to discuss flows in terms of the pressure they exert on a moving body, as this aids the design of many mechanical systems. The two types of pressure that arise in any fluid are hydrostatic and hydrodynamic pressure, the latter of which relates to the flow characteristics of fluid during motion. In this article, we will discuss how these two pressure components arise in real systems and how they can be analyzed from CFD simulation data.
Hydrostatic and Hydrodynamic Pressure
Hydrostatic and hydrodynamic pressure are always present inside a moving fluid and will exert pressure on a nearby body, system wall, or surface of any other object. These two contributions to total pressure exerted by a fluid are exploited in applications like hydraulics, propulsion, and aerodynamics. Each of these pressures is described as follows:
This is the pressure created by the weight of the fluid, or rather due to the force of gravity acting on the fluid. This means a fluid will exert some force on an object depending on the depth of an object into the fluid, regardless of its motion. Hydrostatic pressure has a simple definition based on the depth beneath the fluid surface h, acceleration of gravity g, and fluid density ⍴:
This variation with depth makes sense—the deeper an object is below the surface of a fluid, the more fluid available to exert its weight on the object. Similarly, a denser fluid will have greater weight per unit volume, thus it should exert more pressure for a given depth.
Determining the hydrostatic pressure in any system is relatively simple as long as the density of the fluid is known; simply measure the depth below the surface of the fluid and the hydrostatic pressure is easily calculated. If the fluid is compressible, the same applies as long as the fluid density field can be determined, such as in an FEA/FEM simulation.
This is the pressure a fluid exerts on an object due to the object’s motion through a fluid. Anyone familiar with aerodynamics will likely recognize this as the main contributor to quadratic drag. The hydrodynamic pressure is determined entirely by the motion and density of the fluid as follows:
Just as is the case for hydrostatic pressure, the hydrodynamic pressure can also be determined as long as the density field and velocity field are known. Due to flow behavior in complex systems, CFD simulations are generally needed to calculate these fields. For simpler cases of laminar incompressible inviscid flows, it is possible to identify hydrostatic and hydrodynamic pressure components from analytical fluid dynamics results.
Hydrostatic and Hydrodynamic Pressure in Fluid Dynamics Equations
While it may not be obvious, both hydrostatic and hydrodynamic pressures are present in results that can be derived from the main fluid dynamics equations. Note that the main fluid dynamics equations are statements of conservation of mass and momentum, but we can derive results from which hydrostatic and hydrodynamic pressures can be identified.
For incompressible inviscid laminar flows, hydrostatic and hydrodynamic pressures can be seen in Bernoulli’s equations. Note that this equation defines conservation of mechanical energy per unit mass of fluid, as shown below:
Bernoulli’s equation defines conservation of mechanical energy along every streamline in a laminar flow. By multiplying the constant density across all terms in the above equation, we have another result that applies to every streamline in the flow:
Hydrostatic and hydrodynamic pressures identified in Bernoulli’s equation.
Note that the pressure is included in the hydrodynamic force because the pressure gradient would normally be the main driver of laminar fluid flow.
These two contributions to Bernoulli’s equation match the standard definitions used to describe hydrostatic and hydrodynamic forces, as we outlined above. These results are only applicable to steady incompressible inviscid laminar flows, which are only an approximation of real flow behavior. Unsteady and compressible flows carry different definitions of the forces involved and the resulting pressure acting on a body.
Unsteady and Compressible Flows
In compressible flows, it is possible that mechanical energy is not conserved. This could arise if compression is fast enough for heat to be absorbed or dissipated to the surrounding environment (adiabatic compression/expansion). In this case, the fluid’s mechanical energy would not be conserved and we have an associated heat dissipation problem, possibly involving convection. Bernoulli’s equation would then no longer define a constant along all streamlines.
In the case of isothermal compression/expansion, where compression is slow enough that the fluid temperature does not change over time, mechanical energy will still be conserved and we can redefine the hydrostatic and hydrodynamic pressure components in terms of the adiabatic index:
Hydrostatic and hydrodynamic pressures in Bernoulli’s equation for isothermal compressible laminar inviscid flows.
Determining hydrostatic and hydrodynamic pressure in cases where the fluid is compressible, or when turbulent flow arises, is more mathematically complex. This is compounded by the fact that many fluid dynamics problems can’t be solved by hand in real systems with complex geometry. CFD simulations can be used to determine the hydrostatic and hydrodynamic pressure in a fluid in these more complicated cases.
The first place to start studying hydrostatic and hydrodynamic pressure acting on a complex body is to use a CFD simulation application. When you need to study these effects in complex systems, use the complete set of CFD simulation tools in Omnis from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented with these tools.