$\Delta U=nC_v\Delta T$, is only applicable for ideal gases.
Specific Heat at constant volume i defined as the quantity of heat required to raise the temperature of one mole by $1\text K$,
$$C_v=\frac{\Delta U}{n\Delta T}\tag1$$
This relation is independent of the process that the ideal gas goes through. This follows from the fact that for an ideal gas,w hose particles do not interact which each other, internal energy only depends on the kinetic energy of the particles. Potential internal energy for an ideal gas is $0$.
The only parameter that can change the internal energy of the gas in the kinetic energy of the constituent particles and the only parameter that can change kinetic energy of the particles1 is temperature. Therefore, Internal energy is a function of temperature.
Additional info -
From kinetic theory of gases, Kinetic energy of the particles of an ideal gas only depend on temperature.
For one mole monoatomic ideal gas, $KE=\frac{3}{2}RT$.
From the law of equipartition of energy, the energy associated with each molecule of the system per degree's of freedom of the system is $\frac{1}{2}kT$. For a gas with $f$ degrees of freedom,
$$\text{Total energy associated with each molecule} = \frac{f}{2}kT$$
($k$ is the Boltzmann's constant, $k=\frac{R}{N_a}$).
Similarly,
$$\text{Total energy associated with } n \text{ mole} = \frac{nf}{2}RT$$
Since potential energy of the particles of the system is $0$, The kinetic energy of n mole gas, having $f$ degrees of freedom can be given by,
$$KE=\frac{nf}{2}RT$$
An infinitesimal change in KE will bring an infinitesimal change in internal energy (as stated above),
$dU=dKE$
$dKE=nC_v dt$,
$C_v= \frac{dKE}{dT}$,
$C_v=\frac{nf}{2}R$
Putting this into equation $(1)$,
$$\Delta U=n\frac{f}{2}R \Delta T$ $\tag 2$$
The above relation is more general than the relation tagged $(1)$,
All this calculation assumes no interaction between the particles of the gas, but real gases do interact with each other ,i.e, For real gases potential energy is not $0$.
I hope this adds something to your intuition about internal energy.
Check this out : $\Delta U$at constant temperature , What is the meaning of internal Energy? .
Best Answer
A process where the energy is kept constant is called isoenergetic (or, if you prefer, iso-energetic). It also seems from the literature that a flow where the energy is constant when following a fluid particle is usually called an isoenergetic flow.
Similarly, when the enthalpy is kept constant, the process (or the flow) is said to be isenthalpic (or isoenthalpic). And so on.
Notice that if there is some subtlety and you keep a constant internal energy $U=\text{cte}$ but not a constant energy $E=U+E_{\text{m}}$, by modifying the mechanical energy $E_{\text{m}}$, you should refrain from using standard names like isoenergetic and explain precisely what happens.