Bernoulli's equation normal to streamlines
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Bernoulli's Equation
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today, we will discuss Bernoulli's equation normal to streamlines. Can anyone tell me what we mean by 'normal to the streamlines'?
Does it mean we are looking at points that are perpendicular to the flow direction?
Exactly! Now, does who remember the basic form of Bernoulli's equation we derived earlier?
Is it the one that combines pressure, kinetic, and potential energy?
That's right! We will build on that by adding considerations of curvature and acceleration next.
Deriving Bernoulli's Equation Normal to Streamlines
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let’s derive the equation. We start from the pressure gradient expression.
Why do we use the pressure gradient in the first place?
Great question! The pressure gradient helps us understand how fluid moves due to changes in velocity and elevation. As we develop the equation, we need to consider the centripetal acceleration and gravitational effects as well.
I see, so it's all interconnected!
Exactly! And as a memory aid, remember the acronym VIC: Velocity, Internal pressure, and Curvature. These factors will be crucial in understanding the equation.
Applications of Bernoulli's Equation
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Now, let’s explore some applications such as the stagnation tube. Can anyone explain how this device works?
It measures the flow velocity by using the pressure difference!
Yes! And what about the pitot tube? How is it different?
The pitot tube measures static pressure to determine velocity, right?
Exactly! Both instruments utilize Bernoulli's principle but in slightly different ways. Don't forget the relationships involving pressure differences and flow rates!
Exploring the Venturi Meter
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let’s apply our knowledge to the Venturi meter. How do we use Bernoulli's equation to find flow rates in this system?
We can set up the pressure equations at two points in the meter, right?
Correct! And that leads to understanding the velocity relationship. Can anyone derive that?
Is it because we can relate the velocities to the areas of the sections and their respective pressures?
Yes! Fantastic! Remember this connection as it will be crucial in many applications of fluid mechanics.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Bernoulli's equation normal to streamlines provides insight into fluid behavior under various flow conditions and emphasizes the significance of factors such as pressure differences, curvature, and gravitational effects. It builds on concepts from previous discussions on fluid dynamics to illustrate practical applications.
Detailed
Detailed Summary
In this section, we extend our understanding of Bernoulli's equation by examining its application normal to streamlines. The derivation begins with an acknowledgment of the equation's significance in analyzing fluid motion and forces acting on differential elements of flow. We discuss the force balance in fluid dynamics where the pressure gradient plays a critical role.
The equation is expressed as:
$$\frac{-\Delta p}{\Delta n} + \rho a_n + \rho g \frac{dz}{dn} = 0$$
Here, \(a_n\) represents the centripetal acceleration, given as \(\frac{V^2}{R}\), where \(R\) is the radius of curvature. This relationship indicates the dependence of pressure changes on the curvature of the streamlines. Furthermore, we emphasize that Bernoulli's equation is valid only along a streamline and cannot be crossed without altering the constant terms.
We also review various applications of Bernoulli's equation, such as its use in stagnation tubes and pitot tubes, which are essential for measuring velocities in fluid flows. A detailed example involves the Venturi meter, illustrating how pressure differences can be used to calculate flow rates, reinforcing the practical utility of the Bernoulli equation in engineering applications.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Derivation of the Equation
Chapter 1 of 3
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
So, we have seen this in the previous, you know, when we are dealing with the Bernoulli’s equation along the stream line. So, similarly, we can write minus the force equation we can write, - delta p / delta n + rho a n + rho g dz dn. Here, a n is V square / R, where, R is the local radius of curvature, n is towards the center of the radius of curvature and this is equal to 0, s is constant along the stream line. So, therefore, proceeding in a similar way, we obtain dp = del p del s into ds + delta p delta n into dn, you remember, we had the same along the stream line ds = 0 because and we can write dp dn = del p del n. So, what happens is we get - dp dn = rho V square / R + rho g dz dn because we can write delta p delta n is equal to and we obtained this from this, so, this equation becomes this, the same procedure, no changes whatsoever.
Detailed Explanation
In this section, we begin with the general form of Bernoulli's equation as it relates to the flow of fluids normal to streamlines. We start by noting that the pressure differential can be described by the negative change in pressure over a normal direction (delta p / delta n). This is influenced by the acceleration (a_n), which is derived from the velocity squared divided by the radius of curvature (V²/R). The derivation essentially parallels what was discussed for flow along the streamline, except it accounts for the aspects affecting the flow in the perpendicular direction to the streamline. The result is the equation that relates changes in pressure to velocity and height in a fluid under flow conditions.
Examples & Analogies
Imagine standing next to a fast-flowing river. If you were to throw a small rock into the water, you would notice that the water not only moves downstream but also pushes outward. The pressure changes around the rock as it creates disturbances both directly downstream (along the streamline) and also sideways (normal to the streamline). Just like the analysis of flow normal to the streamlines considers these pressure changes, valuing how the water interacts with the environment around it.
Applications of Bernoulli's Equation
Chapter 2 of 3
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
So, this is the equation we are not going into too much detail about this, but this is how you can have the Bernoulli equation normal to the streamlined. So, you see when you integrate this, this will have a different consequence. So, now the Bernoulli’s equation applications; 1 is stagnation tube that you have already seen, the other was pitot tube which we discussed, the free jets equation we have seen. We have not seen orifice, but orifice is one of the other areas where Bernoulli’s equation is there. Venturi meter, the Bernoulli’s equation is applied across sluice gate and sharp crested weir.
Detailed Explanation
This section addresses the practical applications of Bernoulli’s equation when considering flow normal to streamlines. It lists several fluid mechanics devices that utilize this principle, including stagnation tubes (used in various fluid measurements), pitot tubes (used in aviation for measuring airspeed), and orifices (used for measuring flow rates). The application of the Bernoulli equation in devices like venturi meters demonstrates how engineers can apply theoretical principles to real-world scenarios effectively, providing critical measurements in fluid systems.
Examples & Analogies
Think of a large water slide at a theme park. As you sit at the top with water pooling around you, you are at a point of high pressure but little movement. As you slide down, the angle of the slide changes, creating both forward motion (along the streamline) and pressure that pushes in different directions (normal to the streamlines). The water powering the slide’s flow is similar to applications of Bernoulli’s equation, demonstrating how speed and pressure constantly adjust based on the shape and direction of the slide.
Understanding Flow Behavior
Chapter 3 of 3
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Now, proceeding forward, so, the Bernoulli nation normal to the streamlines. So, can you find the flow Q given the pressure drop between point 1 and 2 and the diameter of the 2 section? You may assume the head loss is negligible. Draw the EGL and the HGL over the contracting section of the venturi metre?
Detailed Explanation
This segment prompts students to apply Bernoulli’s equation in a practical context—calculating flow (Q) using pressure differences measured at two points along a streamline, considering changes in diameter in a venturi meter. By utilizing Bernoulli's principles, students can visualize the hydraulic grade line (HGL) and energy grade line (EGL) over different points in the system. This understanding helps students comprehend how fluids behave under varying pressures and velocities in a fundamental way.
Examples & Analogies
Picture the process of siphoning water from a bucket with a tube. When you suck the air out to start the flow, the initial pressure drop creates a siphon effect, moving water from one bucket to another based on the height difference. The adjustments in speed and pressure along the tube mirror the behavior examined in this section. Calculating flow becomes similar to understanding how the siphon effectively works, reinforcing the principles of fluid mechanics.
Key Concepts
-
Bernoulli's Principle: Describes how energy conservation applies to fluid flows.
-
Pressure Gradient: The change in pressure per unit distance in a fluid.
-
Curvature Effects: How pressure and velocity relate in curved flow paths.
Examples & Applications
Applying Bernoulli's equation in a Venturi meter to calculate flow rates based on pressure differences.
Using pitot tubes to measure air velocity in an aircraft based on the principles of fluid dynamics.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Pressure high, speed low, water flows, where will it go? Curves bend, and force does change, in liquids, it’s all so strange!
Stories
Imagine a river winding through hills. As it bends, the water speeds up on the inside of the curve and slows down on the outside. The pressure changes, shaping the dance of fluid!
Memory Tools
Remember 'PEE' for Bernoulli's: Pressure, Energy, Elevation, the main ingredients of fluid flow.
Acronyms
Use 'PVeG' to recall the crucial factors
Pressure
Velocity
and Gravitational head for Bernoulli's equation.
Flash Cards
Glossary
- Bernoulli's Equation
A principle that describes the relationship between pressure, velocity, and elevation in a fluid flow.
- Streamline
A line that represents the flow of fluids in a steady state, where the flow velocity is tangent to the line.
- Centripetal Acceleration
Acceleration directed towards the center of curvature of a path, relevant to flows in curved streamlines.
Reference links
Supplementary resources to enhance your learning experience.