“Bernoulli's principle applies only to curved surfaces and cannot be applied to flat plate rudders.”

The expressions for rudder lift and drag forces are: FL = 1 2 ρV 2 r SCL, FD = 1 2 ρV 2 r SCD, … Following Hughes et al. (2011), the rudder force model delineates three distinct regimes: pre-stall, transitional, and post-stall, with the latter indicating complete flow detachment. When the angle of attack α is below the stall angle αs, the lift and drag coefficients are: CL = ∂CL ∂α α + CDC ARe α| α| , | α| ≤ αs, CD = CD0 + C2 L πARee0 , | α| ≤ αs.The rudder’s geometry is defined by its root chord cr, tip chord ct, span s and sweep angle of the quarter-chord line Ω. The expressions for rudder lift and drag forces are: FL = 1 2 ρV 2 r SCL, FD = 1 2 ρV 2 r SCD, (50) where S denotes the projected rudder area, Vr represents the average relative flow velocity across the rudder, and CL and CD are the lift and drag coefficients, respectively. Following Hughes et al. (2011), the rudder force model delineates three distinct regimes: pre-stall, transitional, and post-stall, with the latter indicating complete flow detachment. When the angle of attack α is below the stall angle αs, the lift and drag coefficients are: CL = ∂CL ∂α α + CDC ARe α| α| , | α| ≤ αs, CD = CD0 + C2 L πARee0 , | α| ≤ αs. (51)

Furthermore, the explanation based on Bernoulli’s equation along the normal direction to a streamline has a difficulty since both the upper and lower surfaces have zero curvature and a singularity with |∂p/∂n| → ∞ as R → 0 occurs at the leading edge.… For example, a flat-plate airfoil is a clear counterexample for the “equal transit time theory”. The Venturi effect of pinched streamline tubes for higher velocity is not solidly grounded on most upper surface of the plate (except near the leading edge) since the cross section of streamline tubes is expanded there. The application of the Coanda effect is questionable at the sharp leading edge where the flow exhibits a singular behavior. Furthermore, the explanation based on Bernoulli’s equation along the normal direction to a streamline has a difficulty since both the upper and lower surfaces have zero curvature and a singularity with |∂p/∂n| → ∞ as R → 0 occurs at the leading edge. The objective of this paper is to elucidate the physical origin of aerodynamic lift of an airfoil using simple formulations and notations, particularly focusing on the critical effect of the fluid viscosity. The content will be valuable as a supplementary material for teaching in aerodynamics and fluid mechanics. …

Think of a paper airplane. Its airfoil is a flat plate --> top and bottom exactly the same length and shape and yet they fly just fine. … The upper flow is faster and from Bernoulli's equation the pressure is lower. The difference in pressure across the airfoil produces the lift.} As we have seen in Experiment #1, this part of the theory is correct.Let's use the information we've just learned to evaluate the various parts of the "Equal Transit" Theory. - {Lifting airfoils are designed to have the upper surface longer than the bottom.} This is not always correct. The symmetric airfoil in our experiment generates plenty of lift and its upper surface is the same length as the lower surface. Think of a paper airplane. Its airfoil is a flat plate --> top and bottom exactly the same length and shape and yet they fly just fine. This part of the theory probably got started because early airfoils were curved and shaped with a longer distance along the top. Such airfoils do produce a lot of lift and flow turning, but it is the turning that's important, not the distance. There are modern, low-drag airfoils which produce lift on which the bottom surface is actually longer than the top. This theory also does not explain how airplanes can fly upside-down (the longer path would then be on the bottom!) … The lift predicted by the "Equal Transit" theory is much less than the observed lift, because the velocity is too low. The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil. - {The upper flow is faster and from Bernoulli's equation the pressure is lower. The difference in pressure across the airfoil produces the lift.} As we have seen in Experiment #1, this part of the theory is correct. In fact, this theory is very appealing because many parts of the theory are correct. In our discussions on pressure-area integration to determine the force on a body immersed in a fluid, we mentioned that if we know the velocity, we can obtain the pressure and determine the force. The problem with the "Equal Transit" theory is that it attempts to provide us with the velocity based on a non-physical assumption as discussed above.

Bernoulli's equation can be used as a useful tool to calculate a pressure distribution on an airfoil and further airfoil lift when a velocity field around an airfoil is given.Rayleign's lift formula It has been recognized that the working lift models have to be developed based on a cascade of suitable approximations and assumptions since the NS equations are difficult to be directly solved analytically. First, the inviscid-flow approximation is made such that the NS equations are reduced to the Euler equations that have an integral: Bernoulli's equation providing an explicit relation between velocity and pressure along a streamline. Bernoulli's equation can be used as a useful tool to calculate a pressure distribution on an airfoil and further airfoil lift when a velocity field around an airfoil is given. To reconstruct a velocity field around an airfoil, a further approximation is that the flow is potential and irrotational. Therefore, the flow is governed by the Laplace equation of the velocity potential. Some elemental solutions of the Laplace equation can be used as building blocks due to its linear nature to reconstruct a velocity field around an airfoil. In particular, for a two-dimensional (2D) potential flow, a conformal transformation can be applied to the airfoil theory.

Thieme (1965) indicated that the flat-plate profiles may achieve high efficiency in straight-ahead condition. … Liu and Hekkenberg (2015) showed that this high efficiency only appears at small angles of attack, i.e. up to approximately 5◦. After this, the lift coefficient stalls and the lift-to-drag ratio collapses.10 head in Figure 4). Thieme (1965) indicated that the flat-plate profiles may achieve high efficiency in straight-ahead condition. Liu and Hekkenberg (2015) showed that this high efficiency only appears at small angles of attack, i.e. up to approximately 5◦. After this, the lift coefficient stalls and the lift-to-drag ratio collapses. Additionally, flat-plate rudders stall at a smaller angle than other profiles due to earlier and stronger flow separation (Liu and Hekkenberg, 2015). At present, flat-plate rudders are frequently seen on small boats and antique inland vessels but not common for modern seagoing ships. 3.2. NACA NACA profiles are the most widely applied rudder profiles (Kim et al., 2012) (Figure 4). They are also applied to other foil shaped structures such as, propellers Takekoshi et al. …

At low Re, the aerodynamic characteristics of the flat plate are qualitatively similar with those of the NACA0015, and are quantitatively superior to those of the NACA0015.… Moreover, the values of CD of the flat plate approximately coincide with those of the NACA0015. In Fig. 7(c), according to Figs. 7(a) and 7(b), CL/CD tends to increase with increasing α, as well as the NACA0015. Strictly speaking, at α = 12 – 20 deg., CL/CD seems to be almost constant rather than to increase monotonously. In addition, the flat plate achieves a higher value of CL/CD than the NACA0015 at any α in the tested range of α ≤ 20 deg. In summary, at low Re, the aerodynamic characteristics of the flat plate are qualitatively similar with those of the NACA0015, and are quantitatively superior to those of the NACA0015. 3.3 Flow fields around airfoils at low Re Figures 8 and 9 show the flow fields at Re = 1.0×10 2 around the NACA0015 and the flat plate, respectively. Specifically speaking, in each figure, figures (a) and (b) are for α = 4 and 16 deg., respectively. We should note that each distribution is at an instance chosen arbitrarily, because all the tested flows at Re = 1.0×10 2 are completely steady. First, we see Fig. 8. …

Bernoulli's equation relates the pressure on the object to the local velocity; so as the velocity changes around the object, the pressure changes as well. Adding up (integrating) the pressure variation times the area around the entire body determines the aerodynamic force on the body.… Neither Newton nor Bernoulli ever attempted to explain the aerodynamic lift of an object. When a gas flows over an object, or when an object moves through a gas, the molecules of the gas are free to move about the object; they are not closely bound to one another as in a solid. Because the molecules move, there is a velocity (speed plus direction) associated with the gas. Within the gas, the velocity can have very different values at different places near the object. Bernoulli's equation relates the pressure on the object to the local velocity; so as the velocity changes around the object, the pressure changes as well. Adding up (integrating) the pressure variation times the area around the entire body determines the aerodynamic force on the body. The lift is the component of the aerodynamic force which is perpendicular to the original flow direction of the gas. The drag is the component of the aerodynamic force which is parallel to the original flow direction of the gas. Now adding up the velocity variation around the object instead of the pressure variation also determines the aerodynamic force. The integrated velocity variation around the object produces a net turning of the gas flow. …

The existing empirical formula Eq. 2 sets CN∞ = 6.13 sin α and kΛ = 2.25. In this paper, CN∞ of various rudder profiles are obtained through 2D open-water RANS simulations and corrected with the propeller slipstream effects (Table 1)., (3) where kΛ is the rudder aspect ratio impact factor on the normal force coefficient. The existing empirical formula Eq. 2 sets CN∞ = 6.13 sin α and kΛ = 2.25. In this paper, CN∞ of various rudder profiles are obtained through 2D open-water RANS simulations and corrected with the propeller slipstream effects (Table 1). We take the empirical kΛ = 2.25 in terms of the aspect ratio impact factor. A better estimation of kΛ might be achieved through series of 3D RANS simulations. 5. Manoeuvring modelling

JapaneseMeasurement of pressure distributions on a flat plate airfoil were performed using a wind tunnel to examine relationship between flow field and aerodynamics characteristic in low Reynolds number region.Takuya Ono 1, ○Yuta Yamaguchii 1, Yutaro Yamada 1, Tomohisa Ohtake 2, Tatsuo Motohashi 2 Abstract: Measurement of pressure distributions on a flat plate airfoil were performed using a wind tunnel to examine relationship between flow field and aerodynamics characteristic in low Reynolds number region. Validity of pressure coefficients is also confirmed measurement result. Feature of laminar separation bubble on the pressure distribution is observed at an angle of attack of 0 deg, by comparing lift coefficient which calculated by the pressure distribution with the thin airfoil theory, the lift coefficient of 5 deg is differ from the theoretical value at Reynolds number of 20,000 and 70,000. . １． 緒言

With Bernoulli’s equation the velocity is then determined.Figure 12.2: Hot Wire Anemometer [6] 12.3 Hot Wire Anemometer Calibration Before every measurement, the HWA has to be calibrated. During the calibration, the Hot Wire is placed in an uniform flow with known velocity. The flow is generated by blowing through a nozzle with a pressure supply (see Appendix C2). The pressure in the nozzle is measured with a Betz manometer. With Bernoulli’s equation the velocity is then determined. With the StreamWare software, the HWA is then calibrated. It is important that the calibration is performed in the wind tunnel in order for the results to be consistent. Moreover, the probe has to be in the same position (i.e. facing the flow in the same attitude) as it will be in the measurements. 12.4 The Pitot Tube A Pitot tube is also used to measure the flow velocity. By measuring the static and stagnation pressure of the flow, the flow velocity can be obtained.

Consider a steady flow impinging on a perpendicular plate (figure 16). … The fluid along the dividing, or ``stagnation streamline'' slows down and eventually comes to rest without deflection at the stagnation point. Bernoulli's equation along the stagnation streamline givesStagnation pressure and dynamic pressure Bernoulli's equation leads to some interesting conclusions regarding the variation of pressure along a streamline. Consider a steady flow impinging on a perpendicular plate (figure 16). Figure 16. Stagnation point flow. There is one streamline that divides the flow in half: above this streamline all the flow goes over the plate, and below this streamline all the flow goes under the plate. Along this dividing streamline, the fluid moves towards the plate. Since the flow cannot pass through the plate, the fluid must come to rest at the point where it meets the plate. In other words, it ``stagnates.'' The fluid along the dividing, or ``stagnation streamline'' slows down and eventually comes to rest without deflection at the stagnation point. Bernoulli's equation along the stagnation streamline gives where the point e is far upstream and point 0 is at the stagnation point. Since the velocity at the stagnation point is zero,

Situations in which fluid flows at a constant depth are so common that this equation is often also called Bernoulli’s principle, which is simply Bernoulli’s equation for fluids at constant depth.$p1+12ρv12=p2+12ρv22.p1+12ρv12=p2+12ρv22.$ Situations in which fluid flows at a constant depth are so common that this equation is often also called Bernoulli’s principle, which is simply Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) Bernoulli’s principle reinforces the fact that pressure drops as speed increases in a moving fluid: If $v2v2$ is greater than $v1v1$ in the equation, then $p2p2$ must be less than $p1p1$ for the equality to hold. Calculating Pressure

Consider a steady flow impinging on a perpendicular plate (figure 16). … Bernoulli's equation along the stagnation streamline gives where the point e is far upstream and point 0 is at the stagnation point.Stagnation pressure and dynamic pressure Bernoulli's equation leads to some interesting conclusions regarding the variation of pressure along a streamline. Consider a steady flow impinging on a perpendicular plate (figure 16). Figure 16. Stagnation point flow. --- · --- · … Along this dividing streamline, the fluid moves towards the plate. Since the flow cannot pass through the plate, the fluid must come to rest at the point where it meets the plate. In other words, it ``stagnates.'' The fluid along the dividing, or ``stagnation streamline'' slows down and eventually comes to rest without deflection at the stagnation point. Bernoulli's equation along the stagnation streamline gives where the point e is far upstream and point 0 is at the stagnation point. Since the velocity at the stagnation point is zero,