It is probably more proper to start with the NS equations and write all variables as the product of a dimensionful and dimensionless number. For example, the continuity equation becomes, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\rho_0}{t_0}\frac{\partial\rho'}{\partial t'}+\frac{\rho_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0\tag{1}$$ where $\rho=\rho_0\rho'$ with $\rho_0$ the dimensionful quantity and $\rho'$ the dimensionless.

After doing this, you first want to isolate at least one term by multiplying/dividing appropriately (usually you want the highest-order derivative term to have a unit constant). It is at this point that you can then pick your characteristic scales and define the remaining scales as appropriately to satisfy the equations with the goal of making as many coefficients as possible equal to 1.

As an example, we can multiply Eq (1) by $t_0/\rho_0$ to get, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\partial\rho'}{\partial t'}+\frac{t_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0$$ which suggests that if we define any pair of $\ell_0$, $t_0$ and $v_0$ then we can also get the third one for free (using $v_0=\ell_0/t_0$). Note that the density scale is inconsequential in this equation, but almost surely will be defined via the momentum equation.

If you chose to make $v_0=\eta\ell_0/t_0$ for some $\eta\neq1$, then your equations would no longer be identical in form as the dimensionful case (which is the ideal scenario when nondimensionalizing a system of equations): $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\not\leftrightarrow\frac{\partial\rho'}{\partial t'}+\eta\nabla'\cdot\rho'\mathbf{v}'=0$$

So to answer your questions: given Equation 1, you are absolutely constrained to have the velocity scale be the same as the ratio of length and time scales, otherwise your formula are not consistent in the dimensionful and dimensionless cases.

For further reading, see the Wikipedia entry on nondimensionalization. See also the Physics StackExchange posts (both of which I answered) on similar subject matter

• Making Lorentz equation dimensionless for simulations
• Is there a normalized form of the Euler equation discretized with finite volumes?

It is probably more proper to start with the NS equations and write all variables as the product of a dimensionful and dimensionless number. For example, the continuity equation becomes, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\rho_0}{t_0}\frac{\partial\rho'}{\partial t'}+\frac{\rho_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0\tag{1}$$ where $\rho=\rho_0\rho'$ with $\rho_0$ the dimensionful quantity and $\rho'$ the dimensionless.

After doing this, you first want to isolate at least one term by multiplying/dividing appropriately (usually you want the highest-order derivative term to have a unit constant). It is at this point that you can then pick your characteristic scales and define the remaining scales as appropriately to satisfy the equations with the goal of making as many coefficients as possible equal to 1.

As an example, we can multiply Eq (1) by $t_0/\rho_0$ to get, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\partial\rho'}{\partial t'}+\frac{t_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0$$ which suggests that if we define any pair of $\ell_0$, $t_0$ and $v_0$ then we can also get the third one for free (using $v_0=\ell_0/t_0$). Note that the density scale is inconsequential in this equation, but almost surely will be defined via the momentum equation.

If you chose to make $v_0=\eta\ell_0/t_0$ for some $\eta\neq1$, then your equations would no longer be identical in form as the dimensionful case (which is the ideal scenario when nondimensionalizing a system of equations): $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\not\leftrightarrow\frac{\partial\rho'}{\partial t'}+\eta\nabla'\cdot\rho'\mathbf{v}'=0$$

Then you can add to this that there are some variables, such as the pressure scale factor $p_0\sim\rho_0v_0^2$ or the gravity scale factor $g_0\sim v_0/t_0$, that then carry additional multiplicative factors that must be carried about in the momentum equations that make using inconsistent scales even less tenable a scenario.

So to answer your questions: yes, you are absolutely constrained to have the velocity scale be the same as the ratio of length and time scales, otherwise your formula are not consistent in the dimensionful and dimensionless cases.

For further reading, see the Wikipedia entry on nondimensionalization. See also the Physics StackExchange posts (both of which I answered) on similar subject matter

• Making Lorentz equation dimensionless for simulations
• Is there a normalized form of the Euler equation discretized with finite volumes?

It is probably more proper to start with the NS equations and write all variables as the product of a dimensionful and dimensionless number. For example, the continuity equation becomes, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\rho_0}{t_0}\frac{\partial\rho'}{\partial t'}+\frac{\rho_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0\tag{1}$$ where $\rho=\rho_0\rho'$ with $\rho_0$ the dimensionful quantity and $\rho'$ the dimensionless.

After doing this, you first want to isolate at least one term by multiplying/dividing appropriately (usually you want the highest-order derivative term to have a unit constant). It is at this point that you can then pick your characteristic scales and define the remaining scales as appropriately to satisfy the equations with the goal of making as many coefficients as possible equal to 1.

As an example, we can multiply Eq (1) by $t_0/\rho_0$ to get, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\partial\rho'}{\partial t'}+\frac{t_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0$$ which suggests that if we define any pair of $\ell_0$, $t_0$ and $v_0$ then we can also get the third one for free (using $v_0=\ell_0/t_0$). Note that the density scale is inconsequential in this equation, but almost surely will be defined via the momentum equation.

If you chose to make $v_0=\eta\ell_0/t_0$ for some $\eta\neq1$, then your equations would no longer be identical in form as the dimensionful case (which is the ideal scenario): $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\not\leftrightarrow\frac{\partial\rho'}{\partial t'}+\eta\nabla'\cdot\rho'\mathbf{v}'=0$$

So to answer your questions: given Equation 1, you are absolutely constrained to have the velocity scale be the same as the ratio of length and time scales, otherwise your formula are not consistent in the dimensionful and dimensionless cases.

For further reading, see the Wikipedia entry on nondimensionalization. See also the Physics StackExchange posts (both of which I answered) on similar subject matter

• Making Lorentz equation dimensionless for simulations
• Is there a normalized form of the Euler equation discretized with finite volumes?

It is probably more proper to start with the NS equations and write all variables as the product of a dimensionful and dimensionless number. For example, the continuity equation becomes, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\rho_0}{t_0}\frac{\partial\rho'}{\partial t'}+\frac{\rho_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0\tag{1}$$ where $\rho=\rho_0\rho'$ with $\rho_0$ the dimensionful quantity and $\rho'$ the dimensionless.

After doing this, you first want to isolate at least one term by multiplying/dividing appropriately (usually you want the highest-order derivative term to have a unit constant). It is at this point that you can then pick your characteristic scales and define the remaining scales as appropriately to satisfy the equations with the goal of making as many coefficients as possible equal to 1.

As an example, we can multiply Eq (1) by $t_0/\rho_0$ to get, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\to\frac{\partial\rho'}{\partial t'}+\frac{t_0v_0}{\ell_0}\nabla'\cdot\rho'\mathbf{v}'=0$$ which suggests that if we define any pair of $\ell_0$, $t_0$ and $v_0$ then we can also get the third one for free (using $v_0=\ell_0/t_0$). Note that the density scale is inconsequential in this equation, but almost surely will be defined via the momentum equation.

If you chose to make $v_0=\eta\ell_0/t_0$ for some $\eta\neq1$, then your equations would no longer be identical in form as the dimensionful case (which is the ideal scenario when nondimensionalizing a system of equations): $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0\not\leftrightarrow\frac{\partial\rho'}{\partial t'}+\eta\nabla'\cdot\rho'\mathbf{v}'=0$$

So to answer your questions: given Equation 1, you are absolutely constrained to have the velocity scale be the same as the ratio of length and time scales, otherwise your formula are not consistent in the dimensionful and dimensionless cases.

For further reading, see the Wikipedia entry on nondimensionalization. See also the Physics StackExchange posts (both of which I answered) on similar subject matter

• Making Lorentz equation dimensionless for simulations
• Is there a normalized form of the Euler equation discretized with finite volumes?