If $v$ and $w$ are orthogonal smooth vector fields of unit norm and zero divergence, is $\langle w, (\nabla \times v) \times v \rangle=0$?

Question

I am quite confused about manipulating with the curl when the vector field is of unit magnitude and divergence free.

For example, let $v, w : \mathbb{T}^3 \to \mathbb{R}^3$ be periodic smooth vector fields such that $\lVert v(x) \rVert_{L^2}=\lVert w(x) \rVert_{L^2}=1$ for all $x \in \mathbb{R}^3$ and $\nabla \cdot v= \nabla \cdot w=0$.

Here, $\lVert \cdot \rVert_{L^2}$ is the $L^2$ inner product of $L^2[\mathbb{T}^3,\mathbb{R}^3]$.

Moreover, suppose that $\int_{\mathbb{T}^3} v(x) \cdot w(x) d^3x=0$.

Now, I am totally stuck at simplifying the following integral: \begin{equation} \int_{\mathbb{T^3}} w(x) \cdot \bigl( [ \nabla \times v(x)] \times v(x) \bigr) d^3x \end{equation}

Is it possible to do something about the above expression at all? I will delete the question if it is too elementary for MO..