chatgpt-4o：
How do you derive the critical Reynolds number from Navier-Stokes equation?
Deriving the **critical Reynolds number** directly from the **Navier-Stokes equation** is not straightforward. This is because the **Navier-Stokes equation** itself doesn't explicitly predict when a flow will transition from **laminar** to **turbulent**. Instead, the **critical Reynolds number** is typically found empirically or through stability analysis, specifically by applying **linear stability theory** to small perturbations in the laminar flow. The critical Reynolds number corresponds to the point where infinitesimal disturbances in the flow begin to grow, leading to turbulence.
Here’s a step-by-step outline of how the **critical Reynolds number** is determined via stability analysis:
### 1. **Navier-Stokes Equations for Incompressible Flow**
The incompressible Navier-Stokes equations are:
\[
\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla p + \mu \nabla^2 \mathbf{u},
\]
subject to the incompressibility condition:
\[
\nabla \cdot \mathbf{u} = 0.
\]
These equations govern the dynamics of fluid motion. However, they do not inherently indicate the transition from laminar to turbulent flow. To analyze the transition to turbulence, we perform a **stability analysis** on a laminar flow solution.
### 2. **Base Laminar Flow Solution**
We assume there is an initial **steady laminar flow** solution, \( \mathbf{U}(\mathbf{x}) \), to the Navier-Stokes equations. For example, for **pipe flow**, the laminar solution is the **Hagen-Poiseuille flow**, where the velocity profile is parabolic.
For simplicity, we can focus on flows such as:
- **Plane Couette flow** (flow between two parallel plates),
- **Pipe flow** (flow in a circular pipe),
- **Boundary layer flow** (flow over a flat plate).
### 3. **Introduce Small Disturbances**
We introduce small perturbations to the laminar flow. The velocity and pressure fields are perturbed as follows:
\[
\mathbf{u}(\mathbf{x}, t) = \mathbf{U}(\mathbf{x}) + \mathbf{u}'(\mathbf{x}, t),
\]
\[
p(\mathbf{x}, t) = P(\mathbf{x}) + p'(\mathbf{x}, t),
\]
where \( \mathbf{u}' \) and \( p' \) are small perturbations on top of the steady laminar solution \( \mathbf{U}(\mathbf{x}) \) and \( P(\mathbf{x}) \).
### 4. **Linearize the Navier-Stokes Equations**
We substitute these perturbed velocity and pressure fields into the Navier-Stokes equations and subtract out the steady base flow solution. This yields the **linearized Navier-Stokes equations**, which govern the evolution of the small perturbations:
\[
\rho \left( \frac{\partial \mathbf{u}'}{\partial t} + \mathbf{U} \cdot \nabla \mathbf{u}' + \mathbf{u}' \cdot \nabla \mathbf{U} \right) = - \nabla p' + \mu \nabla^2 \mathbf{u}'.
\]
This linearized system can be analyzed to determine whether the small perturbations \( \mathbf{u}' \) grow or decay over time.
### 5. **Normal Mode Analysis**
To investigate the stability of the system, we assume that the perturbations take the form of **normal modes** (i.e., disturbances with a sinusoidal dependence on space and time). For a plane wave perturbation:
\[
\mathbf{u}'(\mathbf{x}, t) = \hat{\mathbf{u}}(y) e^{i(\alpha x - \omega t)},
\]
where:
- \( \alpha \) is the **wave number**,
- \( \omega \) is the **frequency**,
- \( \hat{\mathbf{u}}(y) \) is the amplitude of the perturbation (a function of \( y \)).
Substituting this ansatz into the linearized Navier-Stokes equations gives a set of **ordinary differential equations** for \( \hat{\mathbf{u}}(y) \).
### 6. **Solve the Stability Problem**
The stability problem can now be solved as an **eigenvalue problem**, where \( \omega \) is the eigenvalue. The imaginary part of \( \omega \) determines whether the perturbations grow or decay:
- If \( \text{Im}(\omega) > 0 \), the perturbations grow exponentially with time, leading to instability and transition to turbulence.
- If \( \text{Im}(\omega) < 0 \), the perturbations decay, and the flow remains stable and laminar.
By solving this eigenvalue problem for different Reynolds numbers, we can find the critical Reynolds number \( Re_c \), at which \( \text{Im}(\omega) = 0 \), meaning the flow becomes neutrally stable.
### 7. **Determine the Critical Reynolds Number**
The **critical Reynolds number** \( Re_c \) is the value of the Reynolds number at which the first instability occurs, meaning that at this point, the flow becomes susceptible to the growth of disturbances, leading to turbulence. The critical Reynolds number depends on the flow configuration:
- For **pipe flow**, experimental studies and stability analyses have shown that \( Re_c \approx 2300 \), above which disturbances grow and turbulence begins.
- For **flow over a flat plate** (boundary layer flow), the critical Reynolds number is typically \( Re_x \approx 5 \times 10^5 \), where \( Re_x = \frac{U x}{\nu} \) and \( x \) is the distance from the leading edge.
### Conclusion:
The **critical Reynolds number** is not directly derived from the **Navier-Stokes equation** alone but through a combination of:
1. Identifying the **laminar base flow** solution.
2. Introducing small perturbations and performing a **linear stability analysis**.
3. Solving the **eigenvalue problem** for the perturbations to determine the conditions under which they grow (signaling instability).
The **critical Reynolds number** emerges as the threshold value beyond which the flow becomes unstable, leading to turbulence. This value is different for different flow geometries and must typically be found through theoretical stability analysis or experimental observation.
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FROM 124.14.45.*