Bernoulli Equation

The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy.

Steady-state flow caveat: While the Bernoulli equation is stated in terms of universally valid ideas like conservation of energy and the ideas of pressure, kinetic energy and potential energy, its application in the above form is limited to cases of steady flow. For flow through a tube, such flow can be visualized as laminar flow, which is still an idealization, but if the flow is to a good approximation laminar, then the kinetic energy of flow at any point of the fluid can be modeled and calculated. The kinetic energy per unit volume term in the equation is the one which requires strict constraints for the Bernoulli equation to apply - it basically is the assumption that all the kinetic energy of the fluid is contributing directly to the forward flow process of the fluid. That should make it evident that the existence of turbulence or any chaotic fluid motion would involve some kinetic energy which is not contributing to the advancement of the fluid through the tube.

It should also be said that while conservation of energy always applies, this form of parsing out that energy certainly does not describe how that energy is distributed under transient conditions. A good visualization of the Bernoulli effect is the flow through a constriction, but that neat picture does not describe the fluid when you first turn on the flow.

Another approximation involved in the statement of the Bernoulli equation above is the neglect of losses from fluid friction. Idealized laminar flow through a pipe can be modeled by Poiseuille's law, which does include viscous losses resulting in a lowering of the pressure as you progress along the pipe. The statement of the Bernoulli equation above would lead to the expectation that the pressure would return to the value P₁ past the constriction since the radius returns to its original value. This is not the case because of the loss of some energy from the active flow process by friction into disordered molecular motion (thermal energy). More accurate modeling can be done by combining the Bernoulli equation with Poiseuille's law. A real example which might help visualize the process is the pressure monitoring of the flow through a constricted tube.

Bernoulli Calculation

Now if you can swallow all those assumptions, you can model* the flow in a tube where the volume flowrate is = cm³/s and the fluid density is ρ = gm/cm³. For an inlet tube area A₁= cm² (radius r₁ =cm), the geometry of flow leads to an effective fluid velocity of v₁ =cm/s. Since the Bernoulli equation includes the fluid potential energy as well, the height of the inlet tube is specified as h₁ = cm. If the area of the tube is constricted to A₂=cm² (radius r₂ = cm), then without any further assumptions the effective fluid velocity in the constriction must be v₂ = cm/s. The height of the constricted tube is specified as h₂ = cm.

The kinetic energy densities at the two locations in the tube can now be calculated, and the Bernoulli equation applied to constrain the process to conserve energy, thus giving a value for the pressure in the constriction. First, specify a pressure in the inlet tube:
Inlet pressure = P₁ = kPa = lb/in² = mmHg = atmos.
The energy densities can now be calculated. The energy unit for the CGS units used is the erg.

Inlet tube energy densities 进管能量密度 Kinetic energy density 动能密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：k1。实际数值以上方原输入框为准。 Potential energy density 势能密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：pe1。实际数值以上方原输入框为准。 Pressure energy density 压力能量密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：pr1。实际数值以上方原输入框为准。

Constricted tube energy densities 收缩管的能量密度 Kinetic energy density 动能密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：k2。实际数值以上方原输入框为准。 Potential energy density 势能密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：pe2。实际数值以上方原输入框为准。 Pressure energy density 压力能量密度 = erg/cm3 = erg/cm³ 中文译文中的待填/计算数值依次对应：1：pr2。实际数值以上方原输入框为准。

The pressure energy density in the constricted tube can now be finally converted into more conventional pressure units to see the effect of the constricted flow on the fluid pressure:

Calculated pressure in constriction =
P₂= kPa = lb/in² = mmHg = atmos.

This calculation can give some perspective on the energy involved in fluid flow, but its accuracy is always suspect because of the assumption of laminar flow. For typical inlet conditions, the energy density associated with the pressure will be dominant on the input side; after all, we live at the bottom of an atmospheric sea which contributes a large amount of pressure energy. If a drastic enough reduction in radius is used to yield a pressure in the constriction which is less than atmospheric pressure, there is almost certainly some turbulence involved in the flow into that constriction. Nevertheless, the calculation can show why we can get a significant amount of suction (pressure less than atmospheric) with an "aspirator" on a high pressure faucet. These devices consist of a metal tube of reducing radius with a side tube into the region of constricted radius for suction.

*Note: Some default values will be entered for some of the values as you start exploring the calculation. All of them can be changed as a part of your calculation.

Curve of a Baseball

A non-spinning baseball or a stationary baseball in an airstream exhibits symmetric flow. A baseball which is thrown with spin will curve because one side of the ball will experience a reduced pressure. This is commonly interpreted as an application of the Bernoulli principle and involves the viscosity of the air and the boundary layer of air at the surface of the ball.

The roughness of the ball's surface and the laces on the ball are important! With a perfectly smooth ball you would not get enough interaction with the air. 球的表面粗糙度和球上的绳结都很重要！如果球表面完全光滑，就无法与空气有足够的相互作用。

There are some difficulties with this picture of the curving baseball. The Bernoulli equation cannot really be used to predict the amount of curve of the ball; the flow of the air is compressible, and you can't track the density changes to quantify the change in effective pressure. The experimental work of Watts and Ferrer with baseballs in a wind tunnel suggests another model which gives prominent attention to the spinning boundary layer of air around the baseball. On the side of the ball where the boundary layer is moving in the same direction as the free stream air speed, the boundary layer carries further around the ball before it separates into turbulent flow. On the side where the boundary layer is opposed by the free stream flow, it tends to separate prematurely. This gives a net deflection of the airstream in one direction behind the ball, and therefore a Newton's 3rd law reaction force on the ball in the opposite direction. This gives an effective force in the same direction indicated above.

Similar issues arise in the treatment of a spinning cylinder in an airstream, which has been shown to experience lift. This is the subject of the Kutta-Joukowski theorem. It is also invoked in the discussion of airfoil lift.

Airfoil

The air across the top of a conventional airfoil experiences constricted flow lines and increased air speed relative to the wing. This causes a decrease in pressure on the top according to the Bernoulli equation and provides a lift force. Aerodynamicists (see Eastlake) use the Bernoulli model to correlate with pressure measurements made in wind tunnels, and assert that when pressure measurements are made at multiple locations around the airfoil and summed, they do agree reasonably with the observed lift.

Illustration of lift force and angle of attack 升力力与攻角示意图 Bernoulli vs Newton for airfoil lift 伯努利与牛顿对于机翼升力的对比 Airfoil terminology 机翼术语

Others appeal to a model based on Newton's laws and assert that the main lift comes as a result of the angle of attack. Part of the Newton's law model of part of the lift force involves attachment of the boundary layer of air on the top of the wing with a resulting downwash of air behind the wing. If the wing gives the air a downward force, then by Newton's third law, the wing experiences a force in the opposite direction - a lift. While the "Bernoulli vs Newton" debate continues, Eastlake's position is that they are really equivalent, just different approaches to the same physical phenomenon. NASA has a nice aerodynamics site at which these issues are discussed. 其他人则引用基于牛顿运动定律的模型，并认为升力主要来源于攻角。该模型中的一部分涉及将空气边界层附着在机翼上方，并导致机翼后方空气的下洗现象。如果机翼对空气施加向下的力，那么根据牛顿第三定律，机翼将体验到相反方向的力——即升力。尽管“伯努利与牛顿”争论仍在继续，Eastlake的观点是它们实际上是等效的，只是对同一物理现象的不同方法。NASA有一个很好的空气动力学网站，其中讨论了这些问题。 Increasing the angle of attack gives a larger lift from the upward component of pressure on the bottom of the wing. The lift force can be considered to be a Newton's 3rd law reaction force to the force exerted downward on the air by the wing. 增加攻角会使得机翼下方压力的向上分量产生更大的升力。升力可以被视为机翼对空气施加向下力的牛顿第三定律反作用力。 At too high an angle of attack, turbulent flow increases the drag dramatically and will stall the aircraft. 在攻角过大的情况下，湍流会显著增加阻力并导致飞机失速。

A vapor trail over the wing helps visualize the air flow. Photo by Frank Starmer, used by permission.