How to output a structure sub-navigation of current page
You first have to get the corresponding entry in your "menu" structure and then build your navigation upon that item.

Use the relatedTo() parameter for that. The entry entry model that you automatically get on that single template is the targetElement of that relation, but you should be fine using the less specific element property used with the short syntax relatedTo(myEntryModel).

{# Get the (one!) related entry from the "menu" structure #}
{% set menuItem = craft.entries.section('menu').relatedTo(entry).first() %}

{# Check if there are descendant entries #}
{% if menuItem.hasDescendants() %}

    {# Get entries to build the sub-menu #}
    {% set subMenu = craft.entries.descendantOf(menuItem) %}

    {# Output the sub menu #}
    &lt;ul&gt;
        {% nav link in subMenu %}
            &lt;li&gt;
                &lt;a href="{{ link.url }}"&gt;{{ link.title }}&lt;/a&gt;
                {% ifchildren %}
                    &lt;ul&gt;
                        {% children %}
                    &lt;/ul&gt;
                {% endifchildren %}
            &lt;/li&gt;
        {% endnav %}
    &lt;/ul&gt;

{% endif %}