Math / Algebra

QuestionIf the wind speed at 1 m is $1.5 \mathrm{~ms}^{-1}$ and $4.6 \mathrm{~ms}^{-1}$ at 7 m calculate: (i) The frictional velocity (ii) The roughness parameter $h z_{2}$ (iii) The eddy viscosity coefficient at 5 m (iv) The wind speed at 25 m

Studdy Solution

STEP 1

1. The wind speed at 1 m is $1.5 \, \mathrm{m/s}$ .
2. The wind speed at 7 m is $4.6 \, \mathrm{m/s}$ .
3. We are using the logarithmic wind profile to calculate the required parameters.
4. The von Kármán constant ( $\kappa$ ) is approximately 0.4.

STEP 2

1. Calculate the frictional velocity ( $u_*$ ).
2. Calculate the roughness parameter ( $z_0$ ).
3. Calculate the eddy viscosity coefficient at 5 m.
4. Calculate the wind speed at 25 m.

STEP 3

Calculate the frictional velocity ( $u_$ ) using the logarithmic wind profile equation:
$u(z) = \frac{u_}{\kappa} \ln\left(\frac{z}{z_0}\right)$
We have two equations for two heights:
$1.5 = \frac{u_}{0.4} \ln\left(\frac{1}{z_0}\right)$ $4.6 = \frac{u_}{0.4} \ln\left(\frac{7}{z_0}\right)$
Divide the second equation by the first to eliminate $z_0$ :
$\frac{4.6}{1.5} = \frac{\ln\left(\frac{7}{z_0}\right)}{\ln\left(\frac{1}{z_0}\right)}$
Solve for $u_*$ .

STEP 4

Calculate the roughness parameter ( $z_0$ ).
Use the value of $u_$ found in STEP_1 in either of the original equations to solve for $z_0$ .
$1.5 = \frac{u_}{0.4} \ln\left(\frac{1}{z_0}\right)$
Solve for $z_0$ .

STEP 5

Calculate the eddy viscosity coefficient ( $\nu_t$ ) at 5 m.
The eddy viscosity coefficient is given by:
$\nu_t = \kappa u_* z$
Substitute $z = 5$ m and the value of $u_*$ from STEP_1.

STEP 6

Calculate the wind speed at 25 m using the logarithmic wind profile:
$u(25) = \frac{u_}{0.4} \ln\left(\frac{25}{z_0}\right)$
Substitute the values of $u_$ and $z_0$ obtained in previous steps.

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QuestionIf the wind speed at 1 m is $1.5 \mathrm{~ms}^{-1}$ and $4.6 \mathrm{~ms}^{-1}$ at 7 m calculate: (i) The frictional velocity (ii) The roughness parameter $h z_{2}$ (iii) The eddy viscosity coefficient at 5 m (iv) The wind speed at 25 m

Studdy Solution

STEP 1

1. The wind speed at 1 m is $1.5 \, \mathrm{m/s}$ .
2. The wind speed at 7 m is $4.6 \, \mathrm{m/s}$ .
3. We are using the logarithmic wind profile to calculate the required parameters.
4. The von Kármán constant ( $\kappa$ ) is approximately 0.4.

STEP 2

1. Calculate the frictional velocity ( $u_*$ ).
2. Calculate the roughness parameter ( $z_0$ ).
3. Calculate the eddy viscosity coefficient at 5 m.
4. Calculate the wind speed at 25 m.

STEP 3

STEP 4

Calculate the roughness parameter ( $z_0$ ).
Use the value of $u_$ found in STEP_1 in either of the original equations to solve for $z_0$ .
$1.5 = \frac{u_}{0.4} \ln\left(\frac{1}{z_0}\right)$
Solve for $z_0$ .

STEP 5

Calculate the eddy viscosity coefficient ( $\nu_t$ ) at 5 m.
The eddy viscosity coefficient is given by:
$\nu_t = \kappa u_* z$
Substitute $z = 5$ m and the value of $u_*$ from STEP_1.

STEP 6

Calculate the wind speed at 25 m using the logarithmic wind profile:
$u(25) = \frac{u_}{0.4} \ln\left(\frac{25}{z_0}\right)$
Substitute the values of $u_$ and $z_0$ obtained in previous steps.

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QuestionIf the wind speed at 1 m is 1.5 ms−11.5 \mathrm{~ms}^{-1}1.5 ms−1 and 4.6 ms−14.6 \mathrm{~ms}^{-1}4.6 ms−1 at 7 m calculate: (i) The frictional velocity (ii) The roughness parameter hz2h z_{2}hz2​ (iii) The eddy viscosity coefficient at 5 m (iv) The wind speed at 25 m

Studdy Solution

STEP 1

1. The wind speed at 1 m is 1.5 m/s1.5 \, \mathrm{m/s}1.5m/s.2. The wind speed at 7 m is 4.6 m/s4.6 \, \mathrm{m/s}4.6m/s.3. We are using the logarithmic wind profile to calculate the required parameters.4. The von Kármán constant (κ\kappaκ) is approximately 0.4.

STEP 2

1. Calculate the frictional velocity (u∗u_*u∗​).2. Calculate the roughness parameter (z0z_0z0​).3. Calculate the eddy viscosity coefficient at 5 m.4. Calculate the wind speed at 25 m.

STEP 3

STEP 4

Calculate the roughness parameter (z0z_0z0​).Use the value of u∗u_*u∗​ found in STEP_1 in either of the original equations to solve for z0z_0z0​.1.5=u∗0.4ln⁡(1z0)1.5 = \frac{u_*}{0.4} \ln\left(\frac{1}{z_0}\right)1.5=0.4u∗​​ln(z0​1​)Solve for z0z_0z0​.

STEP 5

Calculate the eddy viscosity coefficient (νt\nu_tνt​) at 5 m.The eddy viscosity coefficient is given by:νt=κu∗z\nu_t = \kappa u_* zνt​=κu∗​zSubstitute z=5z = 5z=5 m and the value of u∗u_*u∗​ from STEP_1.

STEP 6

Calculate the wind speed at 25 m using the logarithmic wind profile:u(25)=u∗0.4ln⁡(25z0)u(25) = \frac{u_*}{0.4} \ln\left(\frac{25}{z_0}\right)u(25)=0.4u∗​​ln(z0​25​)Substitute the values of u∗u_*u∗​ and z0z_0z0​ obtained in previous steps.

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QuestionIf the wind speed at 1 m is $1.5 \mathrm{~ms}^{-1}$ and $4.6 \mathrm{~ms}^{-1}$ at 7 m calculate: (i) The frictional velocity (ii) The roughness parameter $h z_{2}$ (iii) The eddy viscosity coefficient at 5 m (iv) The wind speed at 25 m

1. The wind speed at 1 m is $1.5 \, \mathrm{m/s}$ .
2. The wind speed at 7 m is $4.6 \, \mathrm{m/s}$ .
3. We are using the logarithmic wind profile to calculate the required parameters.
4. The von Kármán constant ( $\kappa$ ) is approximately 0.4.

1. Calculate the frictional velocity ( $u_*$ ).
2. Calculate the roughness parameter ( $z_0$ ).
3. Calculate the eddy viscosity coefficient at 5 m.
4. Calculate the wind speed at 25 m.

Calculate the roughness parameter ( $z_0$ ).
Use the value of $u_$ found in STEP_1 in either of the original equations to solve for $z_0$ .
$1.5 = \frac{u_}{0.4} \ln\left(\frac{1}{z_0}\right)$
Solve for $z_0$ .

Calculate the eddy viscosity coefficient ( $\nu_t$ ) at 5 m.
The eddy viscosity coefficient is given by:
$\nu_t = \kappa u_* z$
Substitute $z = 5$ m and the value of $u_*$ from STEP_1.

Calculate the wind speed at 25 m using the logarithmic wind profile:
$u(25) = \frac{u_}{0.4} \ln\left(\frac{25}{z_0}\right)$
Substitute the values of $u_$ and $z_0$ obtained in previous steps.